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Sobolev norms of local zeta integrals

ABSTRACT

We study the continuity of archimedean zeta integrals, as well as the lower bound of their operator norms, with respect to a natural family of Sobolev norms arising from the functional calculus of harmonic oscillator. The relevant notion and results will be generalized to archimedean Rankin–Selberg integrals of the tempered principal series representations of general linear groups.

1. INTRODUCTION AND MAIN RESULTS

Let F be a number field. Denote by |$\mathbb{A}_F$| the adèle ring of F. Given |$n\in\mathbb{N}$|⁠, let |$\mathrm{GL}_n$| be the general linear group defined over F with degree n. Fix an automorphic cuspidal representation |$(\pi,V_\pi)$| of |$\mathrm{GL}_n(\mathbb{A}_F)$|⁠. Then π decomposes as |$\pi=\widehat{\otimes}^{^{\prime}}\pi_v$|⁠, where |$\widehat{\otimes}^{^{\prime}}$| stands for the completed restricted tensor product, v runs over the set of places of F and |$(\pi_v,V_{\pi_v})$|s are irreducible admissible unitary representations of |$\mathrm{GL}_n(F_v)$|⁠. In this decomposition, π_v is unramified for all but finitely many places. Given any factorizable cusp form |$f\in V_\pi$|⁠, the global |$\mathrm{GL}_n\times\mathrm{GL}_1$| Rankin–Selberg integral (with a trivial character on |$\mathrm{GL}_1$|⁠) splits into local Rankin–Selberg integrals:

$$\Psi(s,f)=\prod_v\Psi_v(s,f_v),\qquad \mathrm{Re}(s)\gg1,\ \,f=\underset{v}{{\otimes}} f_v,\ \,f_v\in V_{\pi_v}.$$

The local L-function |$L(s,\pi_v)$| attached to π_v never takes value 0, and

$$\Psi^\circ_v(s,f_v)=\frac{\Psi_v(s,f_v)}{L(s,\pi_v)}$$

is an entire function on |$\mathbb{C}$| (see [13, Theorem 2.1] for |$v\,|\,{\infty}$|⁠, and [14, Section 2.7] for |$v\nmid{\infty}$|⁠). Let S be the finite set of places v of F, where |$v\,|\,{\infty}$| or, when |$v\nmid{\infty}$|⁠, π_v is ramified. For |$v\notin S$|⁠, we may choose ‘nice’ test vectors f_v so that |$\Psi^\circ_v(s,f_v)\equiv1$| (see [4] and the references therein). Then |$\Psi(s,f)$| is identified with the standard (or, principal) L-function of π, multiplied by the local Rankin–Selberg integrals for |$v\in S$|⁠:

$$ \Psi(s,f)=L(s,\pi)\cdot\prod_{v\in S}\Psi_v(s,f_v),\qquad\mathrm{Re}(s)\gg1. $$

Modulo the convergence and analytic continuation of the quantities in the above identity (which are well known, see [13, 14]), to study the s-aspect properties of |$L(s,\pi)$|⁠, we may focus on the periods |$\Psi(s,\cdot)$| and |$\Psi_v(s,\cdot)$| where |$v\,|\,{\infty}$|⁠, as it is usually easy to handle those |$\Psi_v(s,f_v)$| for the finite places |$v\in S$|⁠; in particular, one may study the operator norms of |$\Psi(s,\cdot)$| and |$\Psi_v(s,\cdot)$|⁠, denoted |$\left\|\Psi(s,\cdot)\right\|_{\text{op}}$| and |$\left\|\Psi_v(s,\cdot)\right\|_{\text{op}}$|⁠, respectively, with respect to prescribed and interrelated Sobolev norms of the representations π and π_v. The expectation is that the variable s occurs in a nice way (ideally, s grows or decays polynomially) in both operator norms, from which we may estimate the growth rate of |$L(s,\pi)$|⁠. For example, fixing the test vectors f_v for |$v\not\in S$|⁠, the upper bound of |$L(s,\pi)$| can be derived from the upper bound of |$\left\|\Psi(s,\cdot)\right\|_{\text{op}}$| and the lower bound of |$\left\|\Psi_v(s,\cdot)\right\|_{\text{op}}$|⁠, where |$v\,|\,{\infty}$|⁠. Nowadays, the (termwise) Sobolev norms of global and local periods have been a powerful tool for studying L-functions (see, for example, [2, 17–19, 21, 22]). In this paper we will deal with the local side: fixing a family of Sobolev norms on principal series representations of |$\mathrm{GL}_n(\mathbb{R})$| and |$\mathrm{GL}_n(\mathbb{C})$|⁠, we study the continuity of the local Rankin–Selberg integrals with respect to these norms, as well as the lower bound of their operator norms.

1.1. The case of |$\mathrm{GL}_1$|

Fix the standard basis of |$\mathfrak{s}\mathfrak{l}_2(\mathbb{R})$|⁠:

$$H=\begin{bmatrix}1&0\\0&-1\end{bmatrix},\quad E^+=\begin{bmatrix}0&1\\0&0\end{bmatrix},\quad E^-=\begin{bmatrix}0&0\\1&0\end{bmatrix}.$$

Then |$\mathfrak{s}\mathfrak{l}_2(\mathbb{R})$| acts on the Schwartz space |$\mathcal{S}(\mathbb{R}^n)$| by

$$\omega(H)=\frac{n}{2}+\sum_{j=1}^nx_j\frac{\partial}{\partial x_j},\quad \omega(E^+)=\frac{\mathbf{i}}{2}\sum_{j=1}^nx_j^2,\quad \omega(E^-)=\frac{\mathbf{i}}{2}\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2},$$

which exponentiates to the oscillator representation of the metaplectic group |$\widetilde{\mathrm{SL}}_2(\mathbb{R})$| on |$L^2(\mathbb{R}^n)$|⁠.

Let |$\mathsf{k}$| be an archimedean local field. Fix a nontrivial unitary additive character ψ of |$\mathsf{k}$|⁠. Denote by |$\mathrm{d} x$| the self-dual Haar measure of |$\mathsf{k}$| associated with ψ. Equip the Schwartz space |$\mathcal{S}(\mathsf{k})$| with the standard inner product with respect to |$\mathrm{d} x$|⁠, denoted |$\left\langle\,,\,\right\rangle_{\mathsf{k}}$|⁠. The harmonic oscillator

$$D_{\mathsf{k}}:=-2\mathbf{i}\cdot\omega(E^+-E^-)=\begin{cases}x^2-\frac{\mathrm{d}^2}{\mathrm{d} x^2},&\text{if} \ \mathsf{k}=\mathbb{R},\\x\bar x-4\partial_x\partial_{\bar x},&\text{if}\ \mathsf{k}=\mathbb{C},\end{cases}$$

is positive and self-adjoint on |$\mathcal{S}(\mathsf{k})$| with respect to |$\left\langle\,,\,\right\rangle_{\mathsf{k}}$|⁠, where we have identified |$\mathbb{C}$| with |$\mathbb{R}^2$|⁠. In addition, there is an orthonormal basis |$\{e_{\mathbf{m}}\}$| of |$\big(\mathcal{S}(\mathsf{k}),\left\langle\,,\,\right\rangle_{\mathsf{k}}\big)$| such that each |$e_{\mathbf{m}}$| is an eigenfunction of |$D_{\mathsf{k}}$|⁠, say, |$D_{\mathsf{k}} =\lambda_{\mathbf{m}}\cdot e_{\mathbf{m}}$|⁠, with |$\lambda_{\mathbf{m}}\in\mathbb{R}_{{\geqslant}1}$|⁠. Here, m is an arbitrary element in |$\mathbb{N}_0$| (for |$\mathsf{k}=\mathbb{R}$|⁠) or in |$\mathbb{N}_0\times\mathbb{N}_0$| (for |$\mathsf{k}=\mathbb{C}$|⁠). We refer the reader to Section 2 for detailed descriptions of |$e_{\mathbf{m}}$| and |$\lambda_{\mathbf{m}}$|⁠. Given any |$\alpha\in\mathbb{R}$|⁠, define |$D_{\mathsf{k}}^\alpha$| on |$\mathcal{S}(\mathsf{k})$| spectrally and formally:

$$D_{\mathsf{k}}^\alpha(e_{\mathbf{m}})=\lambda_{\mathbf{m}}^\alpha\cdot e_{\mathbf{m}}.$$

Then |$D_{\mathsf{k}}^\alpha$| is a well-defined operator on |$\mathcal{S}(\mathsf{k})$| (see Lemma 2.9). Moreover, |$D_{\mathsf{k}}^\alpha$| is positive and self-adjoint with respect to |$\left\langle\,,\,\right\rangle_{\mathsf{k}}$|⁠. The inner product

$$\left\langle f,h\right\rangle_{D_{\mathsf{k}}^\alpha}=\left\langle D_{\mathsf{k}}^\alpha f,h\right\rangle_{\mathsf{k}}=\left\langle D_{\mathsf{k}}^{\frac{\alpha}{2}}f,D_{\mathsf{k}}^{\frac{\alpha}{2}}h\right\rangle_{\mathsf{k}},\qquad f,\ h\in\mathcal{S}(\mathsf{k}),$$

induces the Sobolev norm on |$\mathcal{S}(\mathsf{k})$|⁠:

$$\left\|f\right\|_{D_{\mathsf{k}}^\alpha}=\left\langle f,f\right\rangle_{D_{\mathsf{k}}^\alpha}^{\frac{1}{2}},\qquad f\in\mathcal{S}(\mathsf{k}).$$

The set of Sobolev norms |$\left\|\,\right\|_{D_{\mathsf{k}}^\alpha}$| for |$\alpha{\geqslant} 0$| defines the topology of |$\mathcal{S}(\mathsf{k})$| (see [10, Chapter III, Exercise 8]).

Denote by |$|\,|_{\mathsf{k}}$| the valuation of |$\mathsf{k}$| such that |$\mathrm{d}(ax)=|a|_{\mathsf{k}}\mathrm{d} x$| for any |$a\in\mathsf{k}$|⁠. Then |$\mathrm{d}^\times x=\tfrac{\mathrm{d} x}{|x|_{\mathsf{k}}}$| is a Haar measure of |$\mathsf{k}^\times$|⁠. Fix any continuous unitary character χ of |$\mathsf{k}^\times$|⁠. For |$s\in\mathbb{C}$| with |$\mathrm{Re}(s)\gt0$|⁠, consider the following local zeta integral introduced by Tate in [20]:

$$\begin{array}{rccl} Z_{s,\chi}:&\mathcal{S}(\mathsf{k})&\longrightarrow&\mathbb{C},\\ &f&\longmapsto&\int_{\mathsf{k}^\times}f(x)\chi(x)|x|_{\mathsf{k}}^s\mathrm{d}^\times x.\end{array}$$

Set

$$\theta_{\mathsf{k}}=\left\{\begin{array}{cl}1,&\text{if}\ \mathsf{k}=\mathbb{R},\\ 2,&\text{if}\ \mathsf{k}=\mathbb{C}.\end{array}\right.$$

We will prove the following theorem on the continuity of |$Z_{s,\chi}$| on |$\big(\mathcal{S}(\mathsf{k}),\left\|\,\right\|_{D_{\mathsf{k}}^\alpha}\big)$|⁠.

Theorem 1.1

Assume that |$\mathrm{Re}(s)\in(0,1)$| and |$\alpha\gt\theta_{\mathsf{k}}\cdot|\frac12-\mathrm{Re}(s)|$|⁠. Then |$Z_{s,\chi}$| is continuous on |$\mathcal{S}(\mathsf{k})$| with respect to the Sobolev norm |$\left\|\,\right\|_{D_{\mathsf{k}}^\alpha}$|⁠.

Remark 1.2

The continuity of |$Z_{s,\chi}$| on |$\mathcal{S}(\mathsf{k})$| with respect to the standard L²-norm, which corresponds to α = 0, will also be discussed (see Remark 3.4). We will show that, when χ is trivial, |$Z_{s,\chi}$| fails to be continuous at least for |$s=\frac12$| and 1.

Given |$\mathrm{Re}(s)\in(0,1)$|⁠, Theorem 1.1 implies that |$Z_{s,\chi}$| is continuous on |$\big(\mathcal{S}(\mathsf{k}),\left\|\,\right\|_{D_{\mathsf{k}}^\alpha}\big)$| with α = 1. In this case we will provide a lower bound for the operator norm of |$Z_{s,\chi}$| in terms of s.

Theorem 1.3

Assume that |$\mathrm{Re}(s)\in(0,1)$|⁠. Then there exist positive constants C, Cʹ depending on χ but independent of |$\mathrm{Re}(s)$| such that the operator norm of |$Z_{s,\chi}$| on |$(\mathcal{S}(\mathsf{k}),\|\,\|_{D_{\mathsf{k}}^1})$|satisfies

$$\left\|Z_{s,\chi}\right\|_{\text{op}}{\geqslant} \left\lbrack C\cdot|s|^2+C^{^{\prime}}\right\rbrack^{-\frac12}.$$

In particular,

$$\left\|Z_{s,\chi}\right\|_{\text{op}}\gg \left|s\right|^{-1},\quad\text{as }\,|s|\to{\infty}.$$

Remark 1.4

As |$D_{\mathsf{k}}$| is positive and self-adjoint, it is natural to consider the analogue of the above two theorems associated with other functional calculi of |$D_{\mathsf{k}}$|⁠, like the heat/wave operators. In this respect, our work should be of help.

1.2. The case of |$\mathrm{GL}_n$| with |$n{\geqslant} 2$|

We will generalize the above results on |$Z_{s,\chi}$| to Rankin–Selberg integrals of the tempered principal series representations of |$\mathrm{GL}_n(\mathsf{k})$|⁠. To give an account of them, let us first fix some notations. Denote |$G=\mathrm{GL}_n(\mathsf{k})$| for integer |$n{\geqslant} 2$|⁠, and

A: the set of diagonal matrices of G,
N: the set of unipotent upper triangular matrices of G,
B: the set of upper triangular nonsingular matrices of G,
|$\bar N$|⁠: the opposite of N,
|$\bar B$|⁠: the opposite of B,
ψ: a fixed nontrivial unitary (additive) character of |$\mathsf{k}$|⁠,
|$\mathrm{d} u=\prod\limits_{1{\leqslant} i\lt j{\leqslant} n}\mathrm{d} u_{i,j}$| for |$u=[u_{i,j}]\in N$|⁠: a fixed Haar measure of N,
|$\mathcal{S}(N)$|⁠: the set of Schwartz functions on N.

Any character of A extends to a character of |$\bar B=A\ltimes\bar N$| such that its restriction to |$\bar N$| is trivial. Given characters |$\sigma_1,\cdots,\sigma_n\in\mathrm{Hom}(\mathsf{k}^\times,\mathbb{C}^\times)$|⁠, we have the associated character |$\sigma=\sigma_1{\otimes}\cdots{\otimes}\sigma_n$| of A. Let |$I(\sigma)$| be the set of complex-valued smooth functions on G such that

$$\begin{aligned}f(\bar b g)=\sigma(\bar b)\bar\rho(\bar b)f(g)\end{aligned}$$

for any |$\bar b\in\bar B$| and |$g\in G$|⁠, where |$\bar\rho\in\mathrm{Hom}(A,\mathbb{C}^\times)$| is given by

$$\begin{aligned}\bar\rho:\mathrm{diag}(a_1,\cdots,a_n)\longmapsto\prod_{i=1}^{n}|a_i|_{\mathsf{k}}^{i-1-\frac{n-1}{2}}.\end{aligned}$$

Then the group G acts on |$I(\sigma)$| via the right regular transformations, denoted η_σ. By convention, when there is no confusion arising, we do not distinguish a representation from its underlying space. Set

$$J(\sigma)=\left\lbrace f\in I(\sigma)\colon f|_N\in\mathcal{S}(N)\right\rbrace.$$

When |$I(\sigma)$| is tempered, |$J(\sigma)$| is endowed with the G-invariant inner product

$$ \left\langle f,h\right\rangle=\int_N{f(u)\overline{h(u)}}\mathrm{d} u. $$

(1)

Extend ψ to be a character of N as follows:

$$\psi_N:\,N\longrightarrow\mathbb{C}^\times,\qquad[u_{i,j}]_{1{\leqslant} i\lt j{\leqslant} n}\longmapsto\psi\left(\sum_{i=1}^{n-1}u_{i,i+1}\right).$$

Up to multiplicative scalars, the space |$\mathrm{Hom}_N(I(\sigma),\psi_N)$| has essentially one nonzero element λ. Given |$a\in\mathsf{k}^\times$|⁠, denote

$$[a]=\mathrm{diag}(a,1\cdots,1)\in G.$$

The local Rankin–Selberg integral is then defined (see, for example, [13]) to be

$$Z_s:\,I(\sigma)\longrightarrow\mathbb{C},\qquad f\longmapsto\int_{\mathsf{k}^\times}\lambda([a].f)|a|_{\mathsf{k}}^{s-\frac{n-1}{2}}\mathrm{d}^\times a,$$

which absolutely converges for |$\mathrm{Re}(s)\gt1$|⁠. If |$I(\sigma)$| is tempered, |$Z_s(f)$| absolutely converges for |$\mathrm{Re}(s)\gt0$| and any |$f\in I(\sigma)$| (see [13, Section 5.3]).

For |$1{\leqslant} k\lt l{\leqslant} n$|⁠, set the closed subgroup

$$N_{k,l}=\left\lbrace[u_{i,j}]\in N:u_{i,j}=0\ \,\text{for}\ \,i\lt j\ \,\text{and}\ \,(i,j)\ne(k,l)\right\rbrace.$$

Denote by |$E_{k,l}$| the matrix |$[a_{i,j}]_{1{\leqslant} i,\,j{\leqslant} n}$|⁠, with |$a_{i,j}=1$| for |$(i,j)=(k,l)$|⁠, and |$a_{i,j}=0$| elsewhere. We have the canonical isomorphism

$$\tau_{k,l}:\,\mathsf{k}\longrightarrow N_{k,l},\qquad x\longmapsto\exp\big(xE_{k,l}\big).$$

Denote by |$\tau_{k,l}^\ast$| the pull-back of |$\tau_{k,l}$|⁠, and by |$\tau_{k,l}^\circ$| the push-forward of |$\tau_{k,l}$|⁠. Define the differential operator |$\mathscr{D}_{\mathsf{k}}$| on |$J(\sigma)$| by putting |$D_{\mathsf{k}}$| on each position of |$N_{i,j}$| with i < j:

$$\begin{aligned}(\mathscr{D}_{\mathsf{k}} f)(\bar bu)=\sigma(\bar b)\bar\rho(\bar b)\cdot\prod_{1{\leqslant} i\lt j{\leqslant} n}D_{\mathsf{k}}\big(\tau^\ast_{i,j}(f|_{N_{i,j}})\big)(u_{i,j}),\quad\bar b\in\bar B,\quad u=[u_{i,j}]\in N,\quad f\in J(\sigma),\end{aligned}$$

where |$D_{\mathsf{k}}$| is as before. When |$I(\sigma)$| is tempered, we have an orthonormal basis |$\left\lbrace\mathcal{E}_{\mathbf{M}}\right\rbrace$| of |$\big(J(\sigma),\left\langle\,,\,\right\rangle\big)$|⁠, where

$$\mathcal{E}_{\mathbf{M}}=(\sigma\cdot\overline\rho)|_{\bar B}{\otimes}\Big(\underset{1{\leqslant} i\lt j{\leqslant} n}{{\bigotimes}}\tau_{i,j}^\circ (e_{\mathbf{m}_{i,j}})\Big),\qquad\mathbf{M}=\left(\mathbf{m}_{i,j}\right).$$

Each |$\mathcal{E}_{\mathbf{M}}$| is an eigenfunction of |$\mathscr{D}_{\mathsf{k}}$|⁠:

$$\mathscr{D}_{\mathsf{k}}\left(\mathcal{E}_{\mathbf{M}}\right)=\lambda_{\mathbf{M}}\cdot\mathcal{E}_{\mathbf{M}},$$

with

$$\lambda_{\mathbf{M}}=\prod_{1{\leqslant} i\lt j{\leqslant} n}\lambda_{\mathbf{m}_{i,j}}.$$

For any |$\alpha\in\mathbb{R}$|⁠, define the differential operator |$\mathscr{D}_{\mathsf{k}}^\alpha$| on |$J(\sigma)$| spectrally:

$$\mathscr{D}_{\mathsf{k}}^\alpha\left(\mathcal{E}_{\mathbf{M}}\right)=\lambda_{\mathbf{M}}^\alpha\cdot\mathcal{E}_{\mathbf{M}}.$$

Define

$$\left\langle f,h\right\rangle_{\mathscr{D}_{\mathsf{k}}^\alpha}=\left\langle\mathscr{D}_{\mathsf{k}}^\alpha f,h\right\rangle=\left\langle\mathscr{D}_{\mathsf{k}}^{\frac{\alpha}{2}}f,\mathscr{D}_{\mathsf{k}}^{\frac{\alpha}{2}}h\right\rangle,\qquad f,\ h\in J(\sigma),$$

which induces the Sobolev norm

$$\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}=\left\langle f,f\right\rangle_{\mathscr{D}_{\mathsf{k}}^\alpha}^{\frac{1}{2}},\qquad f\in J(\sigma).$$

When |$I(\sigma)$| is tempered, the functions

$$\mathcal{E}_{\mathbf{M}}^{[\alpha]}:=\lambda_{\mathbf{M}}^{-\frac\alpha2}\mathcal{E}_{\mathbf{M}}$$

constitute an orthonormal basis of |$\big(I(\sigma)_\alpha,\left\langle\,,\,\right\rangle_{\mathscr{D}_{\mathsf{k}}^\alpha}\big)$|⁠, where

$$I(\sigma)_\alpha=\left\lbrace f\in I(\sigma):\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}\lt{\infty}\right\rbrace.$$

We will prove the following theorem on the continuity of Z_s on |$\big(I(\sigma)_\alpha,\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}\big)$|⁠, where α = 1.

Theorem 1.5

Assume that |$I(\sigma)$| is tempered and |$\mathrm{Re}(s)\in(0,1)$|⁠. Then the Rankin–Selberg integral

$$Z_s:\,I(\sigma)_1\longrightarrow\mathbb{C},\qquad f\longmapsto Z_s(f)$$

is continuous with respect to the Sobolev norm |$\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}$|⁠.

For α = 1, we obtain the following lower bound on the operator norm of Z_s.

Theorem 1.6

Assume that |$I(\sigma)$| is tempered and |$\mathrm{Re}(s)\in(0,1)$|⁠. Then there exists a constant C > 0 depending on σ₁ and ψ such that the operator norm of the Rankin–Selberg integral Z_s on |$(I(\sigma)_1,\|\,\|_{\mathscr{D}_{\mathsf{k}}^1})$| satisfies

$$\left\|Z_s\right\|_{\text{op}}{\geqslant} C\cdot \big|\mathrm{Im}(s)\big|^{-1+\theta_{\mathsf{k}}\cdot\left\lbrack\mathrm{Re}(s)-\frac12\right\rbrack},\qquad\text{as}\ \,|\mathrm{Im}(s)|\to{\infty}.$$

In particular, |$\big\|Z_{\frac12+\mathbf{i} t}\big\|_{\text{op}}\gg |t|^{-1}$|⁠, as |$|t|\to{\infty}$|⁠.

Remark 1.7

It is noteworthy that, to prove Theorem 1.5 and 1.6, we do not really work on Z_s. Instead, we shall deal with an orbital integral |$\Lambda_s$| which is an explicit multiple of Z_s (see Section 5). Our experience shows that the orbital integral |$\Lambda_s$| is much easier to handle than Z_s.

Remark 1.8

For |$G=\mathrm{GL}_2(\mathbb{R})$|⁠, we have another type of differential operators and the associated Sobolev norms. Similar properties about Rankin–Selberg integrals are obtained (see Theorem 6.4 and 6.6, and Remark 6.8), which form the content of Section 6.

Notation. In this paper, |$\mathbb{N}_0$| denotes the set of non-negative integers, |$\mathbb{N}$| denotes the set of positive integers, |$\mathbf{i}$| denotes the imaginary unit |$\sqrt{-1}$|⁠, and |$\ll$|⁠, |$\gg$| and |$\asymp$| are Vinogradov symbols.

2. THE SPECTRAL THEORY OF |$D_{\mathsf{k}}$|

Define

$$B_{\mathsf{k}}=\left\{\begin{array}{cl}x-\frac{\mathrm{d}}{\mathrm{d} x},&\text{if}\ \mathsf{k}=\mathbb{R},\\ \frac{\bar x}{2}-\partial_x,&\text{if}\ \mathsf{k}=\mathbb{C}.\end{array}\right.$$

When |$\mathsf{k}=\mathbb{C}$|⁠, we furthermore define

$$\bar B_{\mathbb{C}}=\tfrac{x}{2}-\partial_{\bar x}.$$

The following lemmas can be verified in a straightforward way.

Lemma 2.1

|$B_{\mathbb{C}} \bar B_{\mathbb{C}}=\bar B_{\mathbb{C}} B_{\mathbb{C}}$|⁠.

Lemma 2.2

|$D_{\mathsf{k}} B_{\mathsf{k}}-B_{\mathsf{k}} D_{\mathsf{k}}=2B_{\mathsf{k}}$|⁠.
|$D_{\mathbb{C}} \bar B_{\mathbb{C}}-\bar B_{\mathbb{C}} D_{\mathbb{C}}=2\bar B_{\mathbb{C}}$|⁠.

For |$i\in\mathbb{N}_0$|⁠, denote by H_i the classical real Hermite polynomial given by

$$H_i(x)=(-1)^ie^{x^2}\left(\tfrac{\mathrm{d}}{\mathrm{d} x}\right)^i\big(e^{-x^2}\big),\qquad x\in\mathbb{R}.$$

Clearly, H_i is even when i is even, and odd when i is odd. For |$(i,j)\in\mathbb{N}_0\times\mathbb{N}_0$|⁠, denote by |$H_{i,j}$| the following complex Hermite polynomial introduced by Itô in [12]:

$$ H_{i,j}(x,\bar x)=(-1)^{i+j}e^{x\bar x}\partial_{\bar x}^i\partial_x^j\big(e^{-x\bar x}\big),\qquad x\in\mathbb{C}. $$

(2)

We will write the real/complex Hermite polynomial as |$H_{\mathbf{m}}$| uniformly, where |$\mathbf{m}\in\mathbb{N}_0$| for |$\mathsf{k}=\mathbb{R}$| and |$\mathbf{m}\in\mathbb{N}_0\times\mathbb{N}_0$| for |$\mathsf{k}=\mathbb{C}$|⁠. Set |$\mathfrak{h}_{\mathsf{k}}\in\mathcal{S}(\mathsf{k})$| to be

$$\mathfrak{h}_{\mathsf{k}}(x)=e^{-\frac{x\bar x}{2}},\qquad x\in\mathsf{k}.$$

Define the Hermite function

$$h_{\mathbf{m}}=H_{\mathbf{m}}\cdot \mathfrak{h}_{\mathsf{k}}.$$

Normalize |$h_{\mathbf{m}}$| to be

$$e_{\mathbf{m}}=\left\{\begin{array}{cl}\frac{h_{\mathbf{m}}}{\sqrt{2^ii!\sqrt{\pi}}},&\text{if}\ \mathsf{k}=\mathbb{R}\ \text{and}\ \mathbf{m}=i\in\mathbb{N}_0,\\ \frac{h_{\mathbf{m}}}{\sqrt{\pi i!j!}},&\text{if}\ \mathsf{k}=\mathbb{C}\ \text{and}\ \mathbf{m}=(i,j)\in\mathbb{N}_0\times\mathbb{N}_0.\end{array}\right.$$

Then the Schwartz functions |$e_{\mathbf{m}}$|s constitute an orthonormal basis of |$L^2(\mathsf{k})$| with respect to the standard inner product |$\left\langle\,,\,\right\rangle_{\mathsf{k}}$|⁠. For this fact, one may refer, for example, to [10, Chapter III, Section 2] for the real case and to [7, 11] for the complex case.

The Hermitian functions |$h_{\mathbf{m}}$|s can also be formulated in terms of the operators |$B_{\mathsf{k}}$| and |$\bar B_{\mathbb{C}}$|⁠:

Lemma 2.3

$$ h_{\mathbf{m}}=\left\{\begin{array}{cl}B_{\mathbb{R}}^i(\mathfrak{h}_{\mathsf{k}}),&\text{if}\ \mathsf{k}=\mathbb{R}\ \text{and}\ \mathbf{m}=i\in\mathbb{N}_0,\\ B_{\mathbb{C}}^j \bar B_{\mathbb{C}}^i(\mathfrak{h}_{\mathsf{k}}),&\text{if}\ \mathsf{k}=\mathbb{C}\ \text{and}\ \mathbf{m}=(i,j)\in\mathbb{N}_0\times\mathbb{N}_0.\end{array}\right.$$

Proof.

The real case is treated in [10, Chapter III, Section 2]. Here we prove the complex case. By (2) we have

$$\partial_{\bar x}(H_{i,j})=xH_{i,j}-H_{i+1,j},$$

whence

$$\bar B_{\mathbb{C}}(h_{i,j})=\bar B_{\mathbb{C}}\big(H_{i,j}\cdot e^{-\frac{x\bar x}{2}}\big)=\tfrac{x}{2} h_{i,j}-\left\lbrack\partial_{\bar x}(H_{i,j})-\tfrac{x}{2}H_{i,j}\right\rbrack e^{-\frac{x\bar x}{2}}=H_{i+1,j}e^{-\frac{x\bar x}{2}}=h_{i+1,j}.$$

Similarly, we have

$$B_{\mathbb{C}}(h_{i,j})=h_{i,j+1}.$$

As |$\mathfrak{h}_{\mathsf{k}}=h_{0,0}$|⁠, the formula follows from Lemma 2.1 and an induction on i, j.

Lemma 2.4

|$D_{\mathsf{k}}(h_{\mathbf{m}})=\lambda_{\mathbf{m}}\cdot h_{\mathbf{m}}$|⁠, where

$$\lambda_{\mathbf{m}}=\begin{cases}2i+1,&\text{if}\ \,\mathsf{k}=\mathbb{R}\ \,\text{and}\ \,\mathbf{m}=i\in\mathbb{N}_0,\\ 2(i+j)+2,&\text{if}\ \,\mathsf{k}=\mathbb{C}\ \,\text{and}\ \,\mathbf{m}=(i,j)\in\mathbb{N}_0\times\mathbb{N}_0.\end{cases}$$

Proof.

A simple calculation shows that |$D_{\mathbb{R}}(h_0)=h_0$| and |$D_{\mathbb{C}}(h_{0,0})=2h_{0,0}$|⁠. The rest of the proof is to make induction on m by the use of Lemma 2.2 and Lemma 2.3.

Remark 2.5

|$D_{\mathbb{R}}$| is simply the classical Hermite differential operator.

Remark 2.6

In view of |$\mathcal{S}(\mathbb{C})\simeq\mathcal{S}(\mathbb{R})\widehat{\otimes}\mathcal{S}(\mathbb{R})$|⁠, one may also use |$h_i{\otimes} h_j$|s to describe the spectral decomposition of |$D_{\mathbb{C}}$|⁠.

Lemma 2.7

|$D_{\mathsf{k}}$| is a self-adjoint and positive operator on |$\mathcal{S}(\mathsf{k})$| with respect to the standard inner product.

Proof.

The self-adjointness of |$D_{\mathsf{k}}$| is an immediate consequence of the definition of |$D_{\mathsf{k}}$|⁠. The positivity of |$D_{\mathsf{k}}$| follows from Lemma 2.4.

Lemma 2.8

Write |$f=\sum_{i\in\mathbb{N}_0}b_ie_i\in\mathcal{S}(\mathbb{R})$| with |$b_i\in\mathbb{C}$|⁠. Let |$\mathbb{E}$| be a differential operator on |$\mathbb{R}$| with polynomial coefficients. Then, given any |$\alpha\in\mathbb{R}$|⁠, the series

$$\sum_{i\in\mathbb{N}_0}\lambda_i^\alpha b_i\mathbb{E}(e_i)(x)$$

converges uniformly.

Proof.

Without loss of generality, let us assume throughout the proof that |$\mathbb{E}$| takes the form |$x^m\left(\frac{\mathrm{d}}{\mathrm{d} x}\right)^n$| with m, |$n\in\mathbb{N}_0$|⁠. By definition, |$H_i^{^{\prime}}(x)=2xH_i(x)-H_{i+1}(x)$| for |$i\in\mathbb{N}_0$|⁠. This implies that

$$ e_i^{^{\prime}}(x)=-\sqrt{2(i+1)}\,e_{i+1}+xe_i(x),\qquad i\in\mathbb{N}_0. $$

(3)

By [9, formula 8.952.2], we have

$$ xe_i(x)=\sqrt{\frac{i+1}{2}}\,e_{i+1}(x)+\sqrt{\frac{i}{2}}\,e_{i-1}(x),\qquad i\in\mathbb{N}. $$

(4)

Combining (3) and (4) yields

$$ e^{^{\prime}}_i(x)=-\sqrt{\frac{i+1}{2}}\,e_{i+1}(x)+\sqrt{\frac{i}{2}}\,e_{i-1}(x),\qquad i\in\mathbb{N}. $$

(5)

An iterated calculation based on (4) and (5) shows for |$i{\geqslant} m+n$| that

$$ \mathbb{E}(e_i)(x)=\sum_{k=0}^{m+n}P_k(i)\cdot e_{i-(m+n)+2k}(x), $$

(6)

where P_k is an algebraic function that satisfies

$$ \left|P_k(x)\right|=\left\{\begin{array}{cl}\sqrt{f_{k,k}(x)},&\text{\ for}\ k=0\ \text{or}\ m+n,\\ \sum\limits_{j=1}^{m+n}\sqrt{f_{k,j}(x)},&\text{\ for}\ 0\lt k \lt m+n,\end{array}\right. $$

(7)

with |$f_{k,j}$| being a monic polynomial depending on k and j with degree m + n and taking nonnegative values at |$i{\geqslant} m+n$|⁠. By (6), we have

$$ \sum_{i{\geqslant}2(m+n)}\lambda_i^\alpha b_i\mathbb{E}(e_i)(x)=\sum_{i{\geqslant} m+n}c_ie_i(x), $$

(8)

where

$$c_i=\sum_{k=0}^{m+n}\lambda_{i+m+n-2k}^\alpha b_{i+m+n-2k}P_k(i+m+n-2k).$$

As |$i\to{\infty}$|⁠, the following asymptotics hold:

$$\lambda_{i+m+n-2k}\sim2i\quad\text{and}\quad \left|P_k(i+m+n-2k)\right|\sim \begin{cases}i^{\frac{m+n}{2}},&\text{\ for}\ k=0\ \text{or}\ m+n,\\ (m+n)i^{\frac{m+n}{2}},&\text{\ for}\ 0\lt k \lt m+n.\end{cases}$$

Hence, the series in (8) is majorized by

$$ \sum_{i{\geqslant} m+n}\sum_{k=0}^{m+n}i^{\alpha+\frac{m+n}{2}} b_{i+m+n-2k}\left|e_i(x)\right|. $$

(9)

To prove the lemma, it suffices to show that the series in (9) converges uniformly.

For any |$l\in\mathbb{N}$|⁠, the spectrally defined operator |$D_{\mathbb{R}}^l$| coincides with the l-th power of |$D_{\mathbb{R}}$| in the usual sense and we have

$$D_{\mathbb{R}}^l(f)=\sum_{i\in\mathbb{N}_0}b_i\lambda_i^l e_i.$$

As |$D_{\mathbb{R}}^l(f)$| is a Schwartz function, the inner product

$$\left\langle D_{\mathbb{R}}^l(f),D_{\mathbb{R}}^l(f)\right\rangle_{\mathbb{R}}=\sum_{i\in\mathbb{N}_0}|b_i|^2\lambda_i^{2l}$$

converges for any |$l\in\mathbb{N}$|⁠. This implies that b_i decays faster than any power of i, namely,

$$ b_i\ll_{_{l}}\lambda_i^{-l}\asymp_{_{l}} i^{-l},\qquad\forall\ \,l\in\mathbb{N}. $$

(10)

Now we need the following uniform upper bound of e_i (which follows from [9, formula 8.954.2]):

$$ \max_{x\in\mathbb{R}}|e_i(x)|{\leqslant} C, $$

(11)

where C is a positive constant independent of i. In fact, |$\max_{x\in\mathbb{R}}|e_i(x)|$| decays in i and we have the precise decay rate (see [3, Theorem 1]). But the above bound is enough for our purpose.

Combining (10) and (11) yields the uniform convergence of the series in (9). The lemma is then proved.

Lemma 2.9

|$D_{\mathbb{R}}^\alpha$| is well-defined on |$\mathcal{S}(\mathbb{R})$| for any |$\alpha\in\mathbb{R}$|⁠.

Proof.

Given |$f=\sum_{i\in\mathbb{N}_0}b_ie_i\in\mathcal{S}(\mathbb{R})$|⁠, we need to show for any |$\alpha\in\mathbb{R}$| that

|$D_{\mathbb{R}}^\alpha(f)$| is smooth,
|$D_{\mathbb{R}}^\alpha(f)$| is of rapid decay.

Property (1) follows from Lemma 2.8 by putting |$\mathbb{E}=\frac{\mathrm{d}^n}{\mathrm{d} x^n}$| (⁠|$\forall\ n\in\mathbb{N}$|⁠). It remains to prove (2). We need to show that the series

$$ \mathbb{E}\big(D_{\mathbb{R}}^\alpha(f)\big)(x)=\sum_{i\in\mathbb{N}_0}\lambda_i^\alpha b_i\mathbb{E}(e_i)(x) $$

(12)

is majorized by a constant which is independent of x. For this purpose, since the series (12) converges uniformly by Lemma 2.8, it suffices to show that the partial sum of the series (12) is uniformly bounded, which is true in view of (11).

Lemma 2.10

|$D_{\mathbb{R}}^\alpha$| is bijective on |$\mathcal{S}(\mathbb{R})$|⁠.

Proof.

Note that |$D_{\mathbb{R}}^\alpha\circ D_{\mathbb{R}}^{-\alpha}=\mathrm{id}$| and |$D_{\mathbb{R}}^{-\alpha}\circ D_{\mathbb{R}}^\alpha=\mathrm{id}$|⁠.

Remark 2.11

The complex versions of Lemma 2.8, 2.9 and 2.10 can be proved similarly, noting that |$\left\lbrace e_m{\otimes} e_n\right\rbrace_{m,\,n\in\mathbb{N}_0}$| is an orthogonal basis of |$L^2(\mathbb{C})$| with respect to |$\left\langle\,,\,\right\rangle_{\mathbb{C}}$| and that |$D_{\mathbb{C}}$| is the sum of two |$D_{\mathbb{R}}$|s which are applied to the real variables x and y, respectively, for |$z=x+\mathbf{i} y\in\mathbb{C}$|⁠.

Lemma 2.12

|$D_{\mathsf{k}}^\alpha$| is positive and self-adjoint on |$\mathcal{S}(\mathsf{k})$| with respect to the standard inner product.

Proof.

This is a consequence of the definition of |$D_{\mathsf{k}}^\alpha$| (see Section 1.1) and Lemma 2.7.

Given any |$\alpha\in\mathbb{R}$|⁠, we have the associated inner product |$\left\langle\cdot,\cdot\right\rangle_{D_{\mathsf{k}}^\alpha}$| and Sobolev norm |$\left\|\,\right\|_{D_{\mathsf{k}}^\alpha}$| as defined in Section 1.1. Define

$$e_{\mathbf{m}}^{[\alpha]}=\lambda_{\mathbf{m}}^{-\frac{\alpha}{2}}\cdot e_{\mathbf{m}}.$$

Then |$\{e^{[\alpha]}_{\mathbf{m}}\}\subset\mathcal{S}(\mathsf{k})$| is an orthonormal basis of |$L^2(\mathsf{k})$| with respect to |$\left\langle\,,\,\right\rangle_{D_{\mathsf{k}}^\alpha}$|⁠.

3. THE CONTINUITY OF |$Z_{s,\chi}$|

In this section we will prove Theorem 1.1. Given |$f\in \mathcal{S}(\mathsf{k})$|⁠, write

$$f=\sum_{\mathbf{m}}a_{\mathbf{m}}e_{\mathbf{m}}^{[\alpha]},\qquad a_{\mathbf{m}}\in\mathbb{C},$$

with |$\sum_{\mathbf{m}}|a_{\mathbf{m}}|^2=\|f\|^2_{D_{\mathsf{k}}^\alpha}\lt{\infty}$|⁠. Then

$$\left|Z_{s,\chi}(f)\right|^2=\left|\sum_{\mathbf{m}}a_{\mathbf{m}}Z_{s,\chi}\big(e^{[\alpha]}_{\mathbf{m}}\big)\right|^2{\leqslant}\left(\sum_{\mathbf{m}}|a_{\mathbf{m}}|^2\right)\left(\sum_{\mathbf{m}}\left|Z_{s,\chi}\big(e^{[\alpha]}_{\mathbf{m}}\big)\right|^2\right)=\|f\|^2_{D_{\mathsf{k}}^\alpha}\sum_{\mathbf{m}}\left|Z_{s,\chi}\big(e^{[\alpha]}_{\mathbf{m}}\big)\right|^2.$$

To show that |$Z_{s,\chi}$| is continuous with respect to the Sobolev norm |$\|\,\|_{D_{\mathsf{k}}^\alpha}$|⁠, it suffices to show that |$\sum_{\mathbf{m}}|Z_{s,\chi}(e^{[\alpha]}_{\mathbf{m}})|^2$| is controlled by a constant which might depend on s and α. On the other hand, as an element in the algebraic dual of |$\big(\mathcal{S}(\mathsf{k}),\left\langle\,,\,\right\rangle_{D_{\mathsf{k}}^\alpha}\big)$|⁠, the operator norm |$\left\|Z_{s,\chi}\right\|_{\text{op}}$| of |$Z_{s,\chi}$| is identified with its Hilbert–Schmidt norm, thereby,

$$ \|Z_{s,\chi}\|_{\text{op}}=\sup\limits_{\substack{f\in \mathcal{S}(\mathsf{k})\\ \left\|f\right\|_{_{D_{\mathsf{k}}^\alpha}}=1}}\left|Z_{s,\chi}(f)\right|=\left(\sum_{\mathbf{m}}\left|Z_{s,\chi}( e^{[\alpha]}_{\mathbf{m}})\right|^2\right)^{\frac12}. $$

(13)

3.1. Calculation of |$Z_{s,\chi}(e_{\mathbf{m}}^{[\alpha]})$|

To estimate |$\|Z_{s,\chi}\|_{\text{op}}$|⁠, the first step is to calculate |$Z_{s,\chi}(e^{[\alpha]}_{\mathbf{m}})$|⁠. As χ is a continuous unitary character of |$\mathsf{k}^\times$|⁠, it takes the form:

if |$\mathsf{k}=\mathbb{R}$|⁠, there exists |$z\in\mathbf{i}\mathbb{R}$| such that |$\chi(x)=|x|_{\mathbb{R}}^z$| or |$\mathrm{sgn}(x)\cdot|x|_{\mathbb{R}}^z$| for any |$x\in\mathbb{R}^\times$|⁠;
if |$\mathsf{k}=\mathbb{C}$|⁠, there exist |$\tau\in\mathbb{Z}$| and |$z\in\mathbf{i}\mathbb{R}$| such that |$\chi(x)=r^z e^{i\tau\theta}$| for any |$x=re^{\mathbf{i}\theta}\in\mathbb{C}^\times$|⁠.

3.1.1. The real case

First, we deal with the case where |$\mathsf{k}=\mathbb{R}$|⁠.

Assume that |$\chi(x)=|x|_{\mathbb{R}}^z$| for some |$z\in\mathbf{i}\mathbb{R}$|⁠. By the formula 7.376 of [9], we have for |$\mathrm{Re}(s)\gt0$| that
$$Z_{s,\chi}(h_{\mathbf{m}})=\left\{\begin{array}{cl}0,&\text{if}\ \ \mathbf{m}\in2\mathbb{N}_0+1,\\ (-1)^j\pi^{-\frac12}2^{2j+\frac{s+z}{2}}\Gamma(\tfrac{s+z}{2})\Gamma(j+\tfrac12)F\left(-j,\tfrac{s+z}{2};\tfrac12;2\right),&\text{if }\ \ \mathbf{m}=2j\in2\mathbb{N}_0,\end{array}\right.$$
where F denotes the Gauss hypergeometric function. Hence,
$$ Z_{s,\chi}(e^{[\alpha]}_{\mathbf{m}})=\left\{\begin{array}{cl}0,&\text{if}\ \ \mathbf{m}\in2\mathbb{N}_0+1,\\ \frac{(-1)^j[(2j)!]^{\frac12}2^{-j+\frac{s+z}{2}}}{\pi^{\frac{1}{4}}(4j+1)^{\frac\alpha2}j!}\Gamma(\tfrac{s+z}{2})F\left(-j,\tfrac{s+z}{2};\tfrac12;2\right),&\text{if}\ \ \mathbf{m}=2j\in2\mathbb{N}_0.\end{array}\right. $$
Here and henceforth we use the formula |$\Gamma(j+\frac12)=\frac{(2j)!\sqrt{\pi}}{j!4^j}$|⁠.
Assume that |$\chi(x)=\mathrm{sgn}(x)\cdot|x|_{\mathbb{R}}^z$| for some |$z\in\mathbf{i}\mathbb{R}$|⁠. By the formula 7.376 of [9], we have for |$\mathrm{Re}(s)\gt-1$| that
$$Z_{s,\chi}(h_{\mathbf{m}})=\left\{\begin{array}{cl}0,&\text{if}\ \ \mathbf{m}\in2\mathbb{N}_0,\\ (-1)^j\pi^{-\frac12}2^{2j+3+\frac{s+z-1}{2}}\Gamma(\tfrac{s+z+1}{2})\Gamma(j+\tfrac32)F\left(-j,\tfrac{s+z+1}{2};\tfrac32;2\right),&\text{if}\ \ \mathbf{m}=2j+1\in2\mathbb{N}_0+1.\end{array}\right.$$

Hence,

$$ Z_{s,\chi}\big(e^{[\alpha]}_{\mathbf{m}}\big)=\left\{\begin{array}{cl}0,&\text{if}\ \ \mathbf{m}\in2\mathbb{N}_0,\\ \frac{(-1)^j[(2j+1)!]^{\frac12}2^{1-j+\frac{s+z}{2}}}{\pi^{\frac{1}{4}}(4j+3)^{\frac\alpha2}j!}\Gamma(\tfrac{s+z+1}{2})F\left(-j,\tfrac{s+z+1}{2};\tfrac32;2\right),&\text{if}\ \ \mathbf{m}=2j+1\in2\mathbb{N}_0+1.\end{array}\right. $$

(15)

3.1.2. The complex case

Now we assume that |$\mathsf{k}=\mathbb{C}$| and |$\chi(x)=r^z e^{i\tau\theta}$| for the polar coordinate |$x=re^{\mathbf{i}\theta}$|⁠, where |$\tau\in\mathbb{Z}$| and |$z\in\mathbf{i}\mathbb{R}$|⁠. From the explicit formula

$$H_{i,j}(z,\bar z)=i!j!\sum_{k=0}^{\min(i,\,j)}\frac{(-1)^k}{k!}\frac{z^{i-k}\bar z^{j-k}}{(i-k)!(j-k)!},$$

it follows that

$$H_{i,j}(z,\bar z)=H_{i,j}(r,r)e^{\mathbf{i}(i-j)\theta},\qquad z=re^{\mathbf{i}\theta}.$$

Consequently,

$$Z_{s,\chi}(h_{i,j})=\iint_{[0,{\infty})\times[0,2\pi]}H_{i,j}(r,r)r^{2s+z-1}e^{-\frac{r^2}{2}+\mathbf{i}(i-j+\tau)\theta}\mathrm{d}(r,\theta),$$

which converges absolutely for |$\mathrm{Re}(s)\gt0$|⁠. This, together with the identity |$H_{i,j}(r,r)=H_{j,i}(r,r)$|⁠, implies that

$$\begin{aligned} \overline{Z_{s,\chi}(h_{i,j})}=Z_{\bar s,\bar\chi}(h_{j,i}),\qquad \mathrm{Re}(s)\gt0. \end{aligned}$$

(16)

Moreover, |$Z_{s,\chi}(h_{i,j})=0$| for |$i-j+\tau\ne 0$|⁠. So we assume that |$i-j+\tau=0$| below.

First, let us assume that |$\mathrm{Re}(s)\gt0$| and |$-\tau=i-j{\geqslant} 0$|⁠. Thanks to formula 2.6 of [6], we have
$$h_{i,j}(x)=(-1)^jj!\cdot x^{i-j}L^{(i-j)}_j(x\bar x)\cdot e^{-\frac{x\bar x}{2}},\qquad x\in\mathbb{C},$$
where |$L^{(y)}_j$| denotes the generalized Laguerre polynomial given by
$$L^{(y)}_j(t)=\tfrac{1}{j!}t^{-y}e^t\left(\tfrac{\mathrm{d}}{\mathrm{d} t}\right)^j\big(t^{j+y}e^{-t}\big),\qquad y\gt-1.$$
Using polar coordinate |$x=re^{\mathbf{i}\theta}\in\mathbb{C}$|⁠, one has for |$i{\geqslant} j$| that
$$Z_{s,\chi}(h_{i,j})=(-1)^jj!\iint_{[0,{\infty})\times[0,2\pi]}L_j^{(i-j)}(r^2)r^{(i-j)+2s+z-1}e^{-\frac{r^2}{2}+\mathbf{i}(i-j+\tau)\theta}\mathrm{d}(r,\theta),$$
which converges absolutely for |$\mathrm{Re}(s)\gt0$|⁠. Making the change of variables |$r^2\mapsto x$| and applying [9, formula 7.414.7] yield
$$\begin{aligned} Z_{s,\chi}(h_{i,j})&=(-1)^jj!\pi\int_0^{\infty}L_j^{(-\tau)}(x)e^{-\frac{x}{2}}x^{\frac{-\tau+2s+z}{2}-1}\mathrm{d} x\\ &=(-1)^j\pi\,2^{\frac{-\tau+2s+z}{2}}\frac{\Gamma(\frac{-\tau+2s+z}{2})\Gamma(1-\tau+j)}{\Gamma(1-\tau)}F(-j,\tfrac{-\tau+2s+z}{2};1-\tau;2),\qquad\mathrm{Re}(\tfrac{-\tau+2s}{2})\gt0. \end{aligned}$$
Under our assumption on τ and s, the condition |$\mathrm{Re}(\tfrac{-\tau+2s}{2})\gt0$| is fulfilled and we have for |$i-j+\tau=0$| that
$$ Z_{s,\chi}(e_{i,j}^{[\alpha]})= \frac{(-1)^j\sqrt{\pi}(4j-2\tau+2)^{-\frac\alpha2}(j-\tau)!\,2^{\frac{-\tau+2s+z}{2}}}{(-\tau)!\sqrt{j!\,(j-\tau)!}}\Gamma(\tfrac{-\tau+2s+z}{2})F(-j,\tfrac{-\tau+2s+z}{2};1-\tau;2). $$
Assume that |$\mathrm{Re}(s)\gt0$| and |$\tau=j-i{\geqslant} 0$|⁠. Applying (16) and imitating the calculation in the previous case yield
$$\begin{aligned} Z_{s,\chi}(e_{i,j}^{[\alpha]})&= (-1)^ii!\frac{2\sqrt{\pi}\left(4i+2\tau+2\right)^{-\frac\alpha2}}{\sqrt{i!(i+\tau)!}}\int_0^{\infty}L_i^{(\tau)}(r^2)r^{\tau+2s+z-1}e^{-\frac{r^2}{2}}\mathrm{d} r\nonumber\\ &=(-1)^ii!\frac{\sqrt{\pi}\left(4i+2\tau+2\right)^{-\frac\alpha2}}{\sqrt{i!(i+\tau)!}}\int_0^{\infty}L_i^{(\tau)}(x)e^{-\frac{x}{2}}x^{\frac{\tau+2s+z}{2}-1}\mathrm{d} x\nonumber\\ &=\frac{(-1)^i\sqrt{\pi}\left(4i+2\tau+2\right)^{-\frac\alpha2}(i+\tau)!\,2^{\frac{\tau+2s+z}{2}}}{\tau!\sqrt{i!(i+\tau)!}}\Gamma(\tfrac{\tau+2s+z}{2})F(-i,\tfrac{\tau+2s+z}{2};1+\tau;2). \end{aligned}$$

3.2. Two auxiliary results

We will prove two results to be used in the proof of Theorem 1.1. By Euler’s integral representation of hypergeometric series (see, for example, [1, Theorem 2.2.1]),

$$ F(-j,b;c;2)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}I_j(b-1,c-b-1),\qquad\mathrm{Re}(c)\gt\mathrm{Re}(b)\gt0, $$

(19)

where |$I_j(z_1,z_2)$| is defined below.

Given any |$j\in\mathbb{N}_0$| and z₁, |$z_2\in\mathbb{C}$| such that |$\mathrm{Re}(z_1)\gt-1$| and |$\mathrm{Re}(z_2)\gt-1$|⁠, define the following integral which absolutely converges:

$$I_j(z_1,z_2)=\int_0^1x^{z_1}(1-x)^{z_2}(1-2x)^j\mathrm{d} x.$$

Lemma 3.1

For any ɛ > 0 and z₁, z₂ as above, there exists a constant C > 0 which depends on |$\mathrm{Re}(z_1)$|⁠, |$\mathrm{Re}(z_2)$| and ɛ, such that

$$|I_j(z_1,z_2)|{\leqslant} C\cdot j^{-\min\left\lbrace1+\mathrm{Re}(z_1),\,1+\mathrm{Re}(z_2)\right\rbrace+\varepsilon}$$

for any |$j\in\mathbb{N}$|⁠.

Proof.

First, we have |$\left|I_j(z_1,z_2)\right|{\leqslant} K_j\big(\mathrm{Re}(z_1),\mathrm{Re}(z_2)\big)$|⁠, where

$$K_j(a,b):=\int_0^1 x^a(1-x)^b|1-2x|^j\mathrm{d} x$$

converges for |$a\gt-1,\ b\gt-1$| and |$j\in\mathbb{N}_0$|⁠. Splitting the interval |$[0,1]$| as |$[0,\frac12]\cup[\frac12,1]$| and making the change of variables |$x\mapsto 1-x$| for the integral on |$[\frac12,1]$|⁠, we get

$$ K_j(a,b)=\int_0^{\frac12}x^a(1-x)^b|1-2x|^j\mathrm{d} x+\int_0^{\frac12}x^b(1-x)^a|1-2x|^j\mathrm{d} x=:\mathcal{A}_j(a,b)+\mathcal{A}_j(b,a). $$

Next we estimate |$\mathcal{A}_j(a,b)$|⁠. Fix any |$\beta\in(0,1)$|⁠, we have |$j^\beta\gt2$| for |$j\in\mathbb{N}$| large. Divide the interval |$[0,\frac12]$| into two parts: |$[0,\frac12]=I_1\cup I_2$|⁠, with

$$I_1=[0,\tfrac{1}{j^\beta}],\qquad I_2=[\tfrac{1}{j^\beta},\tfrac12].$$

For any |$x\in I_1$|⁠, if |$b{\geqslant} 0$|⁠, then |$(1-x)^b{\leqslant} 1$|⁠; if |$-1 \lt b\lt0$|⁠, then |$(1-x)^b{\leqslant} \big(\frac12\big)^b\lt2$|⁠. In any case, we have |$x^a(1-x)^b|1-2x|^j{\leqslant} x^a(1-x)^b\lt2x^a$| for |$x\in I_1$|⁠, whence

$$ \int_{I_1}x^a(1-x)^b|1-2x|^j\mathrm{d} x\lt 2\int_{I_1}x^a\mathrm{d} x=\frac{2j^{-(a+1)\beta}}{a+1}. $$

(20)

For any |$x\in I_2$| and large j, we have

$$|1-2x|^j{\leqslant}\left|1-\tfrac{2}{j^\beta}\right|^j=\left|1-\tfrac{2}{j^\beta}\right|^{\left(-\frac{j^\beta}{2}\right)\left(-2j^{1-\beta}\right)}\lt e^{-2j^{1-\beta}},$$

in view that |$\big(1-\frac{2}{j^\beta}\big)^{-\frac{j^\beta}{2}}$| monotunely decreases to e, as j tends to |${\infty}$|⁠. Hence,

$$ \int_{I_2}x^a(1-x)^b|1-2x|^j\mathrm{d} x\lt e^{-2j^{1-\beta}}\int_0^{\frac12}x^a(1-x)^b\mathrm{d} x, $$

(21)

where the integral on the right-hand side converges. Combining (20) and (21) shows that

$$|\mathcal{A}_j(a,b)|\lt \frac{2}{a+1}\cdot j^{-(a+1)\beta}+\check{C}(a,b)\cdot e^{-2j^{1-\beta}},$$

where

$$\check{C}(a,b)=\int_0^{\frac12}x^a(1-x)^b\mathrm{d} x.$$

Switching a and b, we get

$$|\mathcal{A}_j(b,a)|\lt \frac{2}{b+1}\cdot j^{-(1+b)\beta}+\check{C}(b,a)\cdot e^{-2j^{1-\beta}}.$$

Hence,

$$|K_j(a,b)|\lt \frac{4}{\min\{1+a,1+b\}}\cdot j^{-\min\{1+a,1+b\}\beta}+\left(\check{C}(a,b)+\check{C}(b,a)\right)e^{-2j^{1-\beta}}.$$

Substituting |$\beta=1-\frac{\varepsilon}{\min\{1+a,1+b\}}$| yields

$$\left|K_j(a,b)\right|\ll_{_{a,b,\varepsilon}} j^{-\min\{1+a,1+b\}+\varepsilon},$$

from which the lemma follows.

Remark 3.2

It is clear from the proof that the decay of |$I_j(z_1,z_2)$| in j is sensitive to those x near the boundary of |$[0,1]$|⁠. In particular, for any fixed z₁, |$z_2\in\mathbb{R}_{\gt-1}$| and |$\vartheta\in(0,1)$| such that |$j^{-\beta}\in(0,\vartheta)$|⁠, if the integral

$$I_j(\vartheta)=\int_0^\vartheta x^{z_1}(1-x)^{z_2}(1-2x)^j\mathrm{d} x$$

decays polynomially in j, then |$I_j(\vartheta)$| has the same decay order in j with the integral

$$\int_0^{j^{-\beta}}x^{z_1}(1-x)^{z_2}(1-2x)^j\mathrm{d} x,\qquad0\lt \beta\lt 1,$$

since the integral of |$x^{z_1}(1-x)^{z_2}(1-2x)^j$| on |$[j^{-\beta},\vartheta]$| decays exponentially in j.

Given |$j\in\mathbb{N}_0$|⁠, denote

$$a_j=\frac{(2j)!}{4^{\,j}(j\,!)^2}.$$

Lemma 3.3

|$\lim\limits_{j\to{\infty}}\frac{a_j}{j^{-\frac12}}=\frac{1}{\sqrt\pi}$|⁠.

Proof.

As |$\Gamma(j+\frac12)=\frac{(2j)!\sqrt\pi}{j!4^j}$|⁠, we have |$a_j=\frac{\Gamma(j+\frac12)}{j!\sqrt\pi}$|⁠. By Stirling’s asymptotic formula for Gamma function,

$$\frac{\Gamma(j+\frac12)}{j!}=\frac{\Gamma(j+\frac12)}{\Gamma(j+1)}\,\sim\,j^{-\frac12},$$

as |$j\to{\infty}$|⁠. The lemma then follows.

3.3. Proof of Theorem 1.1

First we prove Theorem 1.1 for |$\mathsf{k}=\mathbb{R}$| and |$\chi(x)=|x|_{\mathbb{R}}^z$| with |$z\in\mathbf{i}\mathbb{R}$|⁠. In this case, by (14), (19) and Lemmas 3.1 and 3.3, the convergence of the series in (13) is equivalent to the convergence of

$$\sum_{j\in\mathbb{N}}a_j(4j+1)^{-\alpha}\left|F(-j,\tfrac{s+z}{2};\tfrac12;2)\right|^2\ \asymp\ \sum_{j\in\mathbb{N}}j^{-\frac12-\alpha-\min\{\mathrm{Re}(s),\,1-\mathrm{Re}(s)\}+\varepsilon},$$

where |$\mathrm{Re}(s)\in(0,1)$| so that (14), (19) and Euler’s integral representation of hypergeometric series are valid. The preceding series converges when

$$\alpha\gt\tfrac12-\min\{\mathrm{Re}(s),\,1-\mathrm{Re}(s)\}=\left|\tfrac12-\mathrm{Re}(s)\right|.$$

Next we prove Theorem 1.1 for |$\mathsf{k}=\mathbb{R}$| and |$\chi(x)=\mathrm{sgn}(x)|x|_{\mathbb{R}}^z$|⁠, with |$z\in\mathbf{i}\mathbb{R}$|⁠. In this case, by (15), (19) and Lemma 3.1 and 3.3, the convergence of the series in (13) is equivalent to the convergence of

$$\sum_{j\in\mathbb{N}}a_j(2j+1)(4j+3)^{-\alpha}\left|F(-j,\tfrac{s+z+1}{2};\tfrac32;2)\right|^2\ \asymp\ \sum_{j\in\mathbb{N}}j^{\frac12-\alpha-\min\{1+\mathrm{Re}(s),\,2-\mathrm{Re}(s)\}+\varepsilon},$$

where |$\mathrm{Re}(s)\in(0,2)$| so that |$Z_{s,\chi}(f)$| converges for any |$f\in\mathcal{S}(\mathbb{R})$|⁠; meanwhile (14), (19) and Euler’s integral representation of hypergeometric series are valid. The preceding series converges when

$$\alpha\gt\tfrac32-\min\{1+\mathrm{Re}(s),\,2-\mathrm{Re}(s)\}=\left|\tfrac12-\mathrm{Re}(s)\right|.$$

Finally we prove Theorem 1.1 for |$\mathsf{k}=\mathbb{C}$| and |$\chi(x)=r^ze^{\mathbf{i}\tau\theta}$| for |$z=re^{\mathbf{i}\theta}$|⁠, where |$z\in\mathbf{i}\mathbb{R}$| and |$\tau\in\mathbb{Z}$|⁠. When |$\tau{\leqslant}0$|⁠, by (17), (19) and Lemma 3.1, the convergence of the series in (13) is equivalent to the convergence of

$$ \sum_{j\in\mathbb{N}}(4j-2\tau+2)^{-\alpha}\,\tfrac{(j-\tau)!}{j!}\left|F(-j,\tfrac{-\tau+2s+z}{2};1-\tau;2)\right|^2\ \asymp\ \sum_{j\in\mathbb{N}}j^{-\alpha-\tau-\min\left\lbrace-\tau+2\mathrm{Re}(s),\,2-\tau-2\mathrm{Re}(s)\right\rbrace}. $$

The preceding series converges when

$$\alpha\gt1+\max\left\lbrace-2\mathrm{Re}(s),\,2\mathrm{Re}(s)-2\right\rbrace=\left|1-2\mathrm{Re}(s)\right|.$$

When τ > 0, we may use (18), (19) and Lemma 3.1 to get the same conclusion.

The proof of Theorem 1.1 is then finished.

Remark 3.4

(1) Put α = 0 and |$s=\frac12$|⁠. Let |$\mathsf{k}=\mathbb{R}$| and χ be trivial. By (14),

$$Z_{s,\chi}(e^{[\alpha]}_{\mathbf{m}})=\left\{\begin{array}{cl}0,&\text{if}\ \ \mathbf{m}\in2\mathbb{N}_0+1,\\ \frac{(-1)^j[(2j)!]^{\frac12}2^{-j+\frac{1}{4}}}{\pi^{\frac{1}{4}}j!}\Gamma(\tfrac{1}{4})F\left(-j,\tfrac{1}{4};\tfrac12;2\right),&\text{if}\ \ \mathbf{m}=2j\in2\mathbb{N}_0.\end{array}\right.$$

The convergence of the series (13) is equivalent to that of the series

$$ \sum_{j\in\mathbb{N}}a_j\left|I_j(-\tfrac34,-\tfrac34)\right|^2. $$

(22)

Substituting |$u=\frac12$|⁠, α = 1, λ = 0, |$\nu=\frac{1}{4}$| and |$\mu=j+1$| into formula 3.197 of [9] gives

$$\begin{aligned} \int_0^{\frac12}x^{-\frac34}(1-2x)^j\mathrm{d} x&=2^j\int_0^{\frac12}x^{-\frac34}(\tfrac12-x)^j\mathrm{d} x\nonumber\\ &=2^{-\frac{1}{4}}B(\tfrac14,j+1)F(0,\tfrac14;j+\tfrac54;-\tfrac12)\nonumber\\ &=2^{-\frac{1}{4}}B(\tfrac14,j+1)\nonumber\\ &=2^{-\frac{1}{4}}\frac{\Gamma(\frac{1}{4})\Gamma(j+1)}{\Gamma(j+1+\frac{1}{4})}\nonumber\\ &\sim 2^{-\frac{1}{4}}\Gamma(\tfrac14)j^{-\frac{1}{4}},\qquad \text{as}\ j\to{\infty}, \end{aligned}$$

(23)

where B denotes the beta function, and we have used Stirling’s asymptotic formula for Gamma function in the last step. Since

$$\begin{aligned} I_j(-\tfrac34,-\tfrac34)&=\int_0^{\frac12}t^{-\frac34}(1-t)^{-\frac34}(1-2t)^j\mathrm{d} t+\int_{\frac12}^1t^{-\frac34}(1-t)^{-\frac34}(1-2t)^j\mathrm{d} t\\ &=\left\lbrack1+(-1)^j\right\rbrack\int_0^{\frac12}t^{-\frac34}(1-t)^{-\frac34}(1-2t)^j\mathrm{d} t \end{aligned}$$

(where we have used the the change of variables |$t\mapsto 1-t$| in the second step) and

$$\int_0^{\frac12}t^{-\frac34}(1-2t)^j\mathrm{d} t\ \asymp\ \int_0^{\frac12}t^{-\frac34}(1-t)^{-\frac34}(1-2t)^j\mathrm{d} t,$$

the decay rate of |$I_j(-\frac34,\frac34)$| is |$j^{-\frac{1}{4}}$| by (23). This, together with Lemma 3.3, implies that the series (22) diverges. Hence, |$Z_{\frac12,\mathrm{triv}}$| is not a continuous functional with respect to the standard L²-norm of |$\mathcal{S}(\mathbb{R})$|⁠. Likewise, one can show that |$Z_{\frac12,\mathrm{triv}}$| is not a continuous functional with respect to the standard L²-norm of |$\mathcal{S}(\mathbb{C})$|⁠.

(2) When s = 1 and χ is trivial, the integral representation of hypergeometric series does not hold. In this case, by definition of the hypergeometric series, we simply have

$$F(-j,\tfrac12;\tfrac12;2)=F(-j,1;1;2)=\sum_{\ell=0}^j\left(\substack{j\\ \ell}\right)(-2)^\ell=(-1)^j,$$

By (14), (17) and Lemma 3.1, the series in (13) converges exactly for |$\alpha\gt\frac{\theta_{\mathsf{k}}}{2}$|⁠, whence |$Z_{1,\mathrm{triv}}$| is not a continuous functional on |$\mathcal{S}(\mathsf{k})$| with respect to the standard L²-norm.

Remark 3.5

The square sum of period integrals over a long spectral interval is usually studied via the relative trace formula (see, for example, [8, 16, 23] and the references therein). In view of (13), it is interesting to apply the trace formula method to study |$\left\|Z_{s,\chi}\right\|_{\text{op}}^2$|⁠, despite the obvious difference between our setup and the existing literature.

4. LOWER BOUND OF |$\left\|Z_{s,\chi}\right\|_{\text{op}}$| FOR α = 1

In this section we will prove Theorem 1.3 by using a ‘soft’ argument. The key feature of our result is the polynomial decay in |$|s|$|⁠.

Proof of Theorem 1.3

Let φ be a real-valued non-negative smooth function on |$\mathbb{R}$| whose support is nonempty and lies in (1, 2). For |$b:=\mathrm{Re}(s)\in(0,1)$|⁠, set

$$\varphi_{s,\chi}(x)=|\,|_{\mathsf{k}}^{\bar s}\cdot\bar\chi\cdot\left(\varphi\circ|\,|_{\mathsf{k}}\right).$$

Then |$\varphi_{s,\chi}\in\mathcal{S}(\mathsf{k})$|⁠, and we have

$$Z_{s,\chi}(\varphi_{s,\chi})=\left\{\begin{array}{cl}2\int_0^{\infty} x^{2b}\varphi(x)\mathrm{d}^\times x,&\text{if}\ \ \mathsf{k}=\mathbb{R},\\ 2\pi\int_0^{\infty} x^{4b}\varphi(x^2)\mathrm{d}^\times x,&\text{if}\ \ \mathsf{k}=\mathbb{C},\end{array}\right.$$

where we have used the polar coordinate for |$\mathsf{k}=\mathbb{C}$|⁠. Since the support of φ lies in (1, 2), it follows that |$|x|_{\mathsf{k}}^b{\geqslant} 1$| for |$x\in\mathrm{supp}(\varphi_{s,\chi})$|⁠, whence

$$ Z_{s,\chi}(\varphi_{s,\chi}){\geqslant} 2\int_0^{\infty}\varphi(x)\mathrm{d}^\times x \\[-16pt]$$

(24)

for |$\mathsf{k}=\mathbb{R}$| or |$\mathbb{C}$|⁠, where the integral on the right side is a positive constant.

Under our assumption on φ, the Sobolev norm of |$\varphi_{s,\chi}$| for α = 1 satisfies

if |$\mathsf{k}=\mathbb{R}$| and |$\chi=|\,|_{\mathbb{R}}^z$| or |$\mathrm{sgn}\cdot|\,|_{\mathbb{R}}^z$| with |$z\in\mathbf{i}\mathbb{R}$|⁠, then
$$\begin{aligned} \left\|\varphi_{s,\chi}\right\|^2_{D_{\mathsf{k}}^1} &=\int_{\mathbb{R}} D_{\mathbb{R}}\varphi_{s,\chi}(x)\cdot\overline{\varphi_{s,\chi}(x)}\mathrm{d} x\\ &=\int_{\mathbb{R}}\left|x\varphi_{s,\chi}(x)\right|^2\mathrm{d} x+\int_{\mathbb{R}}\left|\varphi_{s,\chi}^{^{\prime}}(x)\right|^2\mathrm{d} x\\ &{\leqslant}2\int_0^{\infty}|x|^{2+2b}\varphi^2(x)\mathrm{d} x+2\int_0^{\infty}\left|(\bar s+\bar z) x^{\bar s+\bar z-1}\varphi(x)+x^{\bar s+\bar z}\varphi^{^{\prime}}(x)\right|^2\mathrm{d} x\\ &{\leqslant}2\int_0^{\infty}|x|^{2+2b}\varphi^2(x)\mathrm{d} x+4\int_0^{\infty}\left|(\bar s+\bar z) x^{\bar s+\bar z-1}\varphi(x)\right|^2\mathrm{d} x+4\int_0^{\infty}\left|x^{\bar s+\bar z}\varphi^{^{\prime}}(x)\right|^2\mathrm{d} x\\ &{\leqslant}2\int_0^{\infty}|x|^4\varphi^2(x)\mathrm{d} x+4|s+z|^2\int_0^{\infty}\left|\varphi(x)\right|^2\mathrm{d} x+4\int_0^{\infty} x^2|\varphi^{^{\prime}}(x)|^2\mathrm{d} x\\ &=C_1^{^{\prime}}\cdot|s|^2+C_2^{^{\prime}},\end{aligned}$$
where |$C_1^{^{\prime}}$|⁠, |$C_2^{^{\prime}}$| are positive constants depending on φ and χ;
if |$\mathsf{k}=\mathbb{C}$| and |$\chi(x)=r^ze^{\mathbf{i}\tau\theta}$| for |$x=re^{\mathbf{i}\theta}$|⁠, where |$\tau\in\mathbb{Z}$| and |$z\in\mathbf{i}\mathbb{R}$|⁠, then |$\varphi_{s,\chi}(x)=|x|^{\bar s+\frac{\bar z}{2}}\left(\frac{\bar x}{x}\right)^{\frac\tau2}\varphi(x\bar x)$| and an elementary calculation similar to the previous case shows that
$$\begin{aligned} \left\|\varphi_{s,\chi}\right\|^2_{D_{\mathsf{k}}^1} &=\int_{\mathbb{C}} D_{\mathbb{C}}\varphi_{s,\chi}(x)\cdot\overline{\varphi_{s,\chi}(x)}\mathrm{d} x\\ &=\int_{\mathbb{C}}\left|x\varphi_{s,\chi}(x)\right|^2\mathrm{d} x+4\int_{\mathbb{C}}\left|\partial_x\varphi_{s,\chi}(x)\right|^2\mathrm{d} x\\ &{\leqslant} C_1^{^{\prime\prime}}\cdot|s|^2+C_2^{^{\prime\prime}}, \end{aligned}$$
where |$C_1^{^{\prime\prime}}$| and |$C_2^{^{\prime\prime}}$| are positive constants depending on φ and χ.

In any case, we have

$$ \left\|\varphi_{s,\chi}\right\|^2_{D_{\mathsf{k}}^1}{\leqslant} C_1\cdot |s|^2+C_2, $$

(25)

where C₁ and C₂ are positive constants depending on φ and χ. By (24) and (25), there exists a constant C depending on φ such that

$$\frac{\left|Z_{s,\chi}(\varphi_{s,\chi})\right|}{\left\|\varphi_{s,\chi}\right\|_{D_{\mathsf{k}}^1}}{\geqslant} C\cdot\big(C_1\cdot|s|^2+C_2\big)^{-\frac12}.$$

In view that |$\left\|Z_{s,\chi}\right\|_{\text{op}}{\geqslant}\frac{\left|Z_{s,\chi}(\varphi_{s,\chi})\right|}{\left\|\varphi_{s,\chi}\right\|_{D_{\mathsf{k}}^1}}$|⁠, the theorem follows.

Remark 4.1

Inserting specific φ, we get explicit C.

Remark 4.2

It is interesting to find a good upper bound for |$\left\|Z_{s,\chi}\right\|_{\text{op}}$| in terms of s, although this is not our pursuit. Here, by ‘good’ we mean a polynomial bound in |$|s|$| with a small degree.

Remark 4.3

As mentioned in Section 1, it is the estimation of L-functions that motivates our work. However, we cannot handle the global aspect yet.

5. THE CASE OF |$\mathrm{GL}_n(\mathsf{k})$|

In this section, we will extend our work in Sections 2–4 to the tempered principal series representations of |$G=\mathrm{GL}_n(\mathsf{k})$|⁠. We adopt the notations that have appeared in previous sections.

Let R be the first Rankin–Selberg subgroup of G consisting of

$$\begin{bmatrix}y&v\\0&u\end{bmatrix}\in G,$$

where |$y\in\mathsf{k}^\times$|⁠, v is a row vector whose first entry vanishes and |$u\in\mathrm{GL}_{n-1}(\mathsf{k})$| is unipotent upper triangular. Then

$$\mathrm{d}_{\text{r}}g=\mathrm{d}^\times g_{1,1}\cdot\prod\limits_{\substack{1{\leqslant} i\lt j{\leqslant} n\\j\ne 2}}\mathrm{d} g_{i,j},\qquad g=[g_{i,j}]\in R$$

is a right Haar measure of R.

For |$s\in\mathbb{C}$|⁠, define a character ψ_s of R to be

$$\psi_s:\,R\longrightarrow\mathbb{C},\qquad u^{\prime}\cdot[a]\longmapsto|a|_{\mathsf{k}}^{\frac{n-1}{2}-s}\psi_N(u^{\prime}),\qquad u^{\prime}\in R\cap N,\quad a\in\mathsf{k}^\times.$$

Set

$$w_1=\mathrm{diag}\left(\begin{bmatrix}1&1\\0&1\end{bmatrix},1,\cdots,1\right)\in G.$$

Consider the following integral defined in [15]:

$$\Lambda_s(f)=\int_Rf(w_1g)\psi_s^{-1}(g)\mathrm{d}_{\text{r}}g,\qquad f\in I(\sigma).$$

For |$i=1,\cdots,n$|⁠, write |$|\sigma_i(\cdot)|=|\cdot|_{\mathsf{k}}^{\nu_i}$| for some |$\nu_i\in\mathbb{R}$|⁠. Assume that

$$\left\{\begin{array}{c}\max\{\nu_1,\nu_2\}\lt \nu_3\lt \cdots\lt \nu_{n-1}\lt \nu_n,\\\nu_1\lt 1+\nu_2.\end{array}\right.$$

(26)

Then, by [15, Theorem 1.2], |$\Lambda_s(f)$| converges for

$$-\nu_2\lt \mathrm{Re}(s)\lt 1-\nu_1.$$

(27)

If (26) and (27) hold, then both Z_s and |$\Lambda_s$| lie in |$\mathrm{Hom}_R(I(\sigma),\psi_s)$|⁠, and they have the following relation (see Theorem 1.4 of [15]):

$$ \Lambda_s(f)=\gamma(s,\sigma_1,\psi)\cdot Z_s(f), $$

(28)

where γ is the usual local gamma factor.

Fix a nontrivial unitary character ψ of |$\mathsf{k}$| to be

$$\psi(x)=\left\{\begin{array}{cl}e^{\mathbf{i}\mu x},&\text{if}\ \,\mathsf{k}=\mathbb{R},\\ e^{\mathbf{i}(\mu x+\overline{\mu x})},&\text{if}\ \,\mathsf{k}=\mathbb{C},\end{array}\right.$$

where |$\mu\in\mathsf{k}^\times$| is a constant. Define the map

$$T_\psi:\,\mathcal{S}(\mathsf{k})\longrightarrow\mathcal{S}(\mathsf{k}),\qquad f\longmapsto\psi\cdot f.$$

Set

$$c_{\mathbf{m}}=\left\{\begin{array}{cl}\sqrt{(2i+2)\lambda_{i+1}\lambda_i^{-1}},&\text{for }\mathsf{k}=\mathbb{R}\text{ and }\mathbf{m}=i\in\mathbb{N}_0,\\ \sqrt{(i+1)\lambda_{i+1,j}\lambda_{i,j}^{-1}},&\text{for }\mathsf{k}=\mathbb{C}\text{ and }\mathbf{m}=(i,j)\in\mathbb{N}_0\times\mathbb{N}_0.\end{array}\right.$$

When |$\mathsf{k}=\mathbb{C}$|⁠, we furthermore set

$$d_{\mathbf{m}}=\sqrt{(j+1)\lambda_{i,j+1}\lambda_{i,j}^{-1}},\qquad\text{for } \mathbf{m}=(i,j)\in\mathbb{N}_0\times\mathbb{N}_0.$$

Lemma 5.1

(1) Let |$\mathsf{k}=\mathbb{R}$|⁠. For any |$i\in\mathbb{N}$| and |$j\in\mathbb{N}_0$|⁠,

$$\left\langle T_\psi\big(e^{[1]}_i\big),\,T_\psi\big(e^{[1]}_j\big)\right\rangle_{D_{\mathbb{R}}^1}=\left\{\begin{array}{cl}1+\mu^2\lambda_i^{-1},&\text{if }j=i,\\ \mathbf{i}\mu c_i\lambda_{i+1}^{-1},&\text{if }j=i+1,\\ -\mathbf{i}\mu c_{i-1}\lambda_i^{-1},&\text{if }j=i-1,\\ 0,&\text{otherwise}.\end{array}\right.$$

(2) Let |$\mathsf{k}=\mathbb{C}$|⁠. For any |$\mathbf{m}=(i,j)\in\mathbb{N}\times\mathbb{N}$| and |$\mathbf{n}=(r,q)\in\mathbb{N}_0\times\mathbb{N}_0$|⁠,

$$\begin{aligned}\left\langle T_\psi\big(e^{[1]}_{\mathbf{m}}\big),\,T_\psi\big(e^{[1]}_{\mathbf{n}}\big)\right\rangle_{D_{\mathbb{C}}^1}=\left\{\begin{array}{cl} 1+4|\mu|^2\lambda_{\mathbf{m}}^{-1},&\text{if}\ \,(r,q)=(i,j),\\ -2\mathbf{i}\bar{\mu} c_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1},&\text{if}\ \,(r,q)=(i-1,j),\\ 2\mathbf{i}\bar{\mu} d_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1},&\text{if} \ \,(r,q)=(i,j+1),\\ 2\mathbf{i}\mu c_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1},&\text{if}\ \,(r,q)=(i+1,j),\\ -2\mathbf{i}\mu d_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1},&\text{if}\ \,(r,q)=(i,j-1),\\ 0,&otherwise.\end{array}\right.\end{aligned}$$

Proof.

First, we deal with the real case. By definition,

$$D_{\mathbb{R}}\big(T_\psi(f)\big)=T_\psi\big(D_{\mathbb{R}}(f)\big)+\mu^2T_\psi(f)-2\mu\mathbf{i} T_\psi(f^{\prime}),\qquad f\in\mathcal{S}(\mathbb{R}).$$

Hence,

$$\begin{aligned} \left\langle T_\psi\big(e^{[1]}_i\big),\,T_\psi\big(e^{[1]}_j\big)\right\rangle_{D_{\mathbb{R}}^1}&=\left\langle D_{\mathbb{R}}(\psi\cdot e^{[1]}_i),\,\psi\cdot e^{[1]}_j\right\rangle_{\mathbb{R}}\\ &=\left\langle D_{\mathbb{R}} (e_i^{[1]}),\,e_j^{[1]}\right\rangle_{\mathbb{R}}+\mu^2\left\langle e_i^{[1]},\,e_j^{[1]}\right\rangle_{\mathbb{R}}-2\mu\mathbf{i}\left\langle\big(e_i^{[1]}\big)^{\prime},\,e^{[1]}_j\right\rangle_{\mathbb{R}}\\ &=\left\{\begin{array}{cl}\big(\lambda_i+\mu^2\big)\lambda_i^{-1},&\text{if }j=i,\\ -2\mu\mathbf{i}\left\langle\big(e_i^{[1]}\big)^{\prime},\,e^{[1]}_j\right\rangle_{\mathbb{R}},&\text{otherwise},\end{array}\right. \end{aligned}$$

where we have used the fact that |$\big(e^{[1]}_i\big)^{\prime}e^{[1]}_j$| is an odd function on |$\mathbb{R}$| for the case j = i.

In view of Lemma 2.3, we have |$\big(e_i^{[1]}\big)^{\prime}(x)=xe_i^{[1]}-c_ie_{i+1}^{[1]}$|⁠, whence

$$\left\langle\big(e_i^{[1]}\big)^{\prime},\,e^{[1]}_j\right\rangle_{\mathbb{R}}=\left\langle x e_i^{[1]},\,e_j^{[1]}\right\rangle_{\mathbb{R}}-c_i\left\langle e_{i+1}^{[1]},\,e^{[1]}_j\right\rangle_{\mathbb{R}}.$$

On the other hand, integration by parts gives

$$\left\langle\big(e_i^{[1]}\big)^{\prime},\,e^{[1]}_j\right\rangle_{\mathbb{R}}=-\left\langle e^{[1]}_i,\,\big(e^{[1]}_j\big)^{\prime}\right\rangle_{\mathbb{R}}=-\left\langle x e^{[1]}_i,\,e^{[1]}_j\right\rangle_{\mathbb{R}}+c_j\left\langle e^{[1]}_i,\,e^{[1]}_{j+1}\right\rangle_{\mathbb{R}}.$$

Combining these formulas, we get for |$j\ne i$| that

$$\begin{aligned} \left\langle xe^{[1]}_i,\,e^{[1]}_j\right\rangle_{\mathbb{R}}&=\frac12\left\lbrack c_i\left\langle e^{[1]}_{i+1},\,e^{[1]}_j\right\rangle_{\mathbb{R}}+c_j\left\langle e^{[1]}_i,\,e^{[1]}_{j+1}\right\rangle_{\mathbb{R}}\right\rbrack\\ &=\left\{\begin{array}{cl} \frac{c_i}{2\lambda_{i+1}},&\text{if }j=i+1,\\ \frac{c_{i-1}}{2\lambda_i},&\text{if }j=i-1,\\ 0,&\text{otherwise}. \end{array}\right. \end{aligned}$$

The conclusion for the real case then follows.

Now we deal with the complex case using the analogous argument. By definition,

$$\begin{aligned}D_{\mathbb{C}}\big(T_\psi(f)\big)=T_\psi\big(D_{\mathbb{C}}(f)\big)+4|\mu|^2T_\psi(f)-4\mathbf{i}\,\bar\mu T_\psi\big(\partial_x(f)\big)-4\mathbf{i}\mu T_\psi\big(\partial_{\bar x}(f)\big),\qquad f\in\mathcal{S}(\mathbb{C}).\end{aligned}$$

Consequently,

$$\begin{aligned} &\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}=\left\langle D_{\mathbb{C}} \big(T_\psi\big(e_{\mathbf{m}}^{[1]}\big)\big),\,T_\psi\big(e_{\mathbf{n}}^{[1]}\big)\right\rangle_{\mathbb{C}}\\ & \quad =\left\langle D_{\mathbb{C}}(e_{\mathbf{m}}^{[1]}),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}+4|\mu|^2\left\langle e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}-4\mathbf{i}\,\bar{\mu}\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}-4\mathbf{i}\mu\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}\\ & \quad =\left\{\begin{array}{cl}\left(\lambda_{\mathbf{m}}+4|\mu|^2\right)\lambda_{\mathbf{m}}^{-1}-4\mathbf{i}\,\bar\mu\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{m}}^{[1]}\right\rangle_{\mathbb{C}}-4\mathbf{i}\mu\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{m}}^{[1]}\right\rangle_{\mathbb{C}},&\text{if}\ \,\mathbf{m}=\mathbf{n},\\ -4\mathbf{i}\,\bar\mu\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}-4\mathbf{i}\mu\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}},&\text{if}\ \,\mathbf{m}\ne\mathbf{n}, \end{array}\right. \end{aligned}$$

where the last step follows from the basic properties of |$e^{[\alpha]}_{\mathbf{m}}$|s introduced in Section 2.

Set |$\mathbf{m}=(i,j)$| and |$\mathbf{n}=(r,q)$|⁠. In view of Lemma 2.3, we have

$$\begin{aligned} \partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big)(x)&=\tfrac{x}{2}e_{\mathbf{m}}^{[1]}(x)-c_{\mathbf{m}}e_{i+1,j}^{[1]}(x),\\ \partial_x\big(e_{\mathbf{m}}^{[1]}\big)(x)&=\tfrac{\bar x}{2}e_{\mathbf{m}}^{[1]}(x)-d_{\mathbf{m}}e_{i,j+1}^{[1]}(x). \end{aligned}$$

So we may express |$\left\langle\partial_x(e_{\mathbf{m}}^{[1]}),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}$| in two ways. On the one hand,

$$\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=-\left\langle e_{\mathbf{m}}^{[1]},\,\partial_{\bar x}\big(e_{\mathbf{n}}^{[1]}\big)\right\rangle_{\mathbb{C}}=-\left\langle\tfrac{\bar x}{2}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}+c_{\mathbf{n}}\left\langle e_{\mathbf{m}}^{[1]},\,e_{r+1,q}^{[1]}\right\rangle_{\mathbb{C}},$$

where the first step is integration by parts. On the other hand,

$$ \left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=\left\langle\tfrac{\bar x}{2}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}-d_{\mathbf{m}}\left\langle e_{i,j+1}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}. $$

(29)

The two expressions immediately imply that

$$ \left\langle\bar{x}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=c_{\mathbf{n}}\left\langle e_{\mathbf{m}}^{[1]},\,e_{r+1,q}^{[1]}\right\rangle_{\mathbb{C}}+d_{\mathbf{m}}\left\langle e_{i,j+1}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}. $$

(30)

Similarly, we have two expressions for |$\left\langle\partial_{\bar x}(e_{\mathbf{m}}^{[1]}),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}$|⁠:

$$ \left\langle\tfrac{x}{2}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}-c_{\mathbf{m}}\left\langle e_{i+1,j}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=-\left\langle\tfrac{x}{2}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}+d_{\mathbf{n}}\left\langle e_{\mathbf{m}}^{[1]},\,e_{r,q+1}^{[1]}\right\rangle_{\mathbb{C}}, $$

(31)

which yields

$$ \left\langle x e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=c_{\mathbf{m}}\left\langle e_{i+1,j}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}+d_{\mathbf{n}}\left\langle e_{\mathbf{m}}^{[1]},\,e_{r,q+1}^{[1]}\right\rangle_{\mathbb{C}}. $$

(32)

There are six cases to treat separately.

If |$(r,q)=(i,j)$|⁠, then |$\left\langle\bar{x}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=\left\langle x e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠, thanks to (30) and (32), respectively. In view of (29) and (31), this implies that |$\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{m}}^{[1]}\right\rangle_{\mathbb{C}}=\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{m}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠. As a result,
$$\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{m}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}=\big(\lambda_{\mathbf{m}}+4|\mu|^2\big)\lambda_{\mathbf{m}}^{-1}.$$
If |$(r,q)=(i-1,j)$|⁠, then |$\left\langle\bar{x}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=c_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1}$| and |$\left\langle x e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠, which implies that |$\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=\frac12c_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1}$| and |$\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠. As a result,
$$\begin{aligned}\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}=-2\mathbf{i}\,\bar\mu c_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1}.\end{aligned}$$
If |$(r,q)=(i,j+1)$|⁠, then |$\left\langle\bar{x}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=d_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1}$| and |$\left\langle x e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠, which implies that |$\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=-\tfrac12d_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1}$| and |$\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠. As a result,
$$\begin{aligned}\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}=2\mathbf{i}\,\bar\mu d_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1}.\end{aligned}$$
If |$(r,q)=(i+1,j)$|⁠, then |$\left\langle\bar{x}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$| and |$\left\langle x e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=c_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1}$|⁠, which implies that |$\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$| and |$\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=-\tfrac12c_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1}$|⁠. As a result,
$$\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}=2\mathbf{i}\mu c_{\mathbf{m}}\lambda_{\mathbf{n}}^{-1}.$$
If |$(r,q)=(i,j-1)$|⁠, then |$\left\langle\bar{x}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$| and |$\left\langle x e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=d_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1}$|⁠, which implies that |$\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$| and |$\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=\tfrac12d_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1}$|⁠. As a result,
$$\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}=-2\mathbf{i}\mu d_{\mathbf{n}}\lambda_{\mathbf{m}}^{-1}.$$
If none of the above five cases happens, then |$\left\langle\bar{x}e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$| and |$\left\langle x e_{\mathbf{m}}^{[1]},\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠, which implies that |$\left\langle\partial_x\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=\left\langle\partial_{\bar x}\big(e_{\mathbf{m}}^{[1]}\big),\,e_{\mathbf{n}}^{[1]}\right\rangle_{\mathbb{C}}=0$|⁠. As a result,
$$\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}=0.$$

This proves the conclusion for the complex case.

Given any |$\mathbf{m}\in\mathbb{N}_0$| or |$\mathbb{N}_0\times\mathbb{N}_0$|⁠, define

$$\mathbf{m}^\ast=\left\lbrace\mathbf{n}\in\mathbb{N}_0\ \,\text{or}\ \,\mathbb{N}_0\times\mathbb{N}_0:\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathsf{k}}^1}\ne 0\right\rbrace.$$

The following result is a simple consequence of Lemma 5.1.

Corollary 5.2

$|\mathbf{m}^\ast|{\leqslant}\left\{\begin{array}{cl}3,&\text{if}\ \,\mathsf{k}=\mathbb{R},\\ 5,&\text{if}\ \,\mathsf{k}=\mathbb{C}.\end{array}\right.$
⁠.
There exists a constant |$C_\mu\gt0$| depending on µ such that
$$\left|\left\langle T_\psi(e_{\mathbf{m}}^{[1]}),\,T_\psi(e_{\mathbf{n}}^{[1]})\right\rangle_{D_{\mathbb{C}}^1}\right|{\leqslant} C_\mu,$$
for any m and any |$\mathbf{n}\in\mathbf{m}^\ast$|⁠.

Proposition 5.3

The map T_ψ is a continuous bijection on |$\big(\mathcal{S}(\mathsf{k}),\|\,\|_{D_{\mathsf{k}}^1}\big)$| with continuous inverse.

Proof.

That T_ψ is bijective is clear. To show the continuity, it suffices to show that |$\left\|T_\psi(f)\right\|_{D_{\mathsf{k}}^1}$| is bounded by an absolute constant for any |$f=\sum_{\mathbf{m}}a_{\mathbf{m}}e_{\mathbf{m}}^{[1]}\in\mathcal{S}(\mathsf{k})$| such that

$$\left\|f\right\|^2_{D_{\mathsf{k}}^1}=\sum_{\mathbf{m}}|a_{\mathbf{m}}|^2=1,\qquad a_{\mathbf{m}}\in\mathbb{C}.$$

By definition,

$$\begin{aligned} \left\langle T_\psi(f),T_\psi(f)\right\rangle_{D_{\mathsf{k}}^1}&=\sum_{\mathbf{m},\,\mathbf{n}}a_{\mathbf{m}}\bar a_{\mathbf{n}}\left\langle T_\psi(e^{[1]}_{\mathbf{m}}),T_\psi(e^{[1]}_{\mathbf{n}})\right\rangle_{D_{\mathsf{k}}^1}. \end{aligned}$$

We then have

$$\begin{aligned} \left|\sum_{\mathbf{m},\,\mathbf{n}}a_{\mathbf{m}}\bar a_{\mathbf{n}}\left\langle T_\psi(e^{[1]}_{\mathbf{m}}),T_\psi(e^{[1]}_{\mathbf{n}})\right\rangle_{D_{\mathsf{k}}^1}\right|&{\leqslant}\sum_{\mathbf{m}}\sum_{\mathbf{n}\in\mathbf{m}^\ast}\left|a_{\mathbf{m}}\bar a_{\mathbf{n}}\right|\left|\left\langle T_\psi(e^{[1]}_{\mathbf{m}}),T_\psi(e^{[1]}_{\mathbf{n}})\right\rangle_{D_{\mathsf{k}}^1}\right|\notag\\ &{\leqslant} C_\mu\sum_{\mathbf{m}}\sum_{\mathbf{n}\in\mathbf{m}^\ast}\frac{|a_{\mathbf{m}}|^2+|a_{\mathbf{n}}|^2}{2}\notag\\ &{\leqslant} C_\mu\sum_{\mathbf{m}}|\mathbf{m}^\ast|\cdot\frac{|a_{\mathbf{m}}|^2}{2}\\ &{\leqslant} \tfrac{5C_\mu}{2}\sum_{\mathbf{m}}|a_{\mathbf{m}}|^2\notag\\ &=\tfrac{5C_\mu}{2},\notag \end{aligned}$$

(33)

where (33) is due to the fact that each term |$a_{\mathbf{m}}$| in the series occurs |$|\mathbf{m}^\ast|$| times. This proves the continuity of T_ψ. That |$T_\psi^{-1}=T_{\psi^{-1}}$| is continuous is equally treated, which implies that T_ψ has continuous inverse.

Set

$$\begin{aligned}J_0(\sigma)=(\sigma\cdot\bar\rho)|_{\bar B}{\otimes}\left(\underset{1{\leqslant} i\lt j{\leqslant} n}{{\bigotimes}}\mathcal{S}(N_{i,j})\right)\subset J(\sigma).\end{aligned}$$

In view that |$\mathcal{S}(N)=\widehat{\bigotimes}_{1{\leqslant} i\lt j{\leqslant} n}\mathcal{S}(N_{i, j})$|⁠, we may define the map

$$\begin{aligned} \mathcal{T}_\psi:J_0(\sigma)&\xrightarrow J_0(\sigma),\\ (\sigma\cdot\overline\rho)|_{\bar B}{\otimes}\left(\underset{1{\leqslant} i\lt j{\leqslant} n}{{\bigotimes}}\phi_{i,j}\right)&\longmapsto(\sigma\cdot\bar\rho)|_{\bar B}{\otimes}\left(\underset{\substack{j=i+1=2\\ \text{or}\\ j \g ti+1{\geqslant} 2}}{{\bigotimes}}\phi_{i,j}\right){\otimes}\left(\underset{j=i+1{\geqslant} 3}{{\bigotimes}}T_\psi(\phi_{i,j})\right),\quad\phi_{i,j}\in\mathcal{S}(N_{i,j}), \end{aligned}$$

and extend it to |$J(\sigma)$|⁠.

Proposition 5.4

Assume that |$I(\sigma)$| is tempered. Then the map |$\mathcal{T}_\psi$| is a continuous bijection on |$\big(J(\sigma),\|\,\|_{\mathscr{D}_{\mathsf{k}}^1}\big)$| with continuous inverse.

Proof.

This is an immediate consequence of Proposition 5.3.

Lemma 5.5

|$I(\sigma)_\alpha$| is a subspace of |$I(\sigma)$| containing |$J(\sigma)$|⁠.

Proof.

Given f, |$h\in J(\sigma)_\alpha$|⁠, we have

$$\begin{aligned} \left\langle f+h,f+h\right\rangle_{\mathscr{D}_{\mathsf{k}}^\alpha}&=\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}^2+\left\|h\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}^2+\left\langle\mathscr{D}_{\mathsf{k}}^\alpha(f),h\right\rangle+\left\langle\mathscr{D}_{\mathsf{k}}^\alpha(h),f\right\rangle\\ &=\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}^2+\left\|h\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}^2+\left\langle\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(f),\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(h)\right\rangle+\left\langle\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(h),\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(f)\right\rangle\\ &\lt{\infty}, \end{aligned}$$

since

$$\left|\left\langle\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(f),\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(h)\right\rangle\right|^2{\leqslant}\left\langle\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(f),\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(f)\right\rangle\left\langle\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(h),\mathscr{D}_{\mathsf{k}}^{\frac\alpha2}(h)\right\rangle=\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}^2\left\|h\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}^2\lt {\infty}.$$

So |$I(\sigma)_\alpha$| is a subspace of |$I(\sigma)$|⁠.

For any |$l\in\mathbb{N}_0$| and |$f\in J(\sigma)$|⁠, we have |$\mathscr{D}_{\mathsf{k}}^l(f)\in J(\sigma)$| since |$D_{\mathsf{k}}^l$| applied to a Schwartz function always yields a Schwartz function. This implies that if we write |$f\in J(\sigma)$| as |$f=\sum_{\mathbf{M}}a_{\mathbf{M}}\mathcal{E}_{\mathbf{M}}$|⁠, where |$a_{\mathbf{M}}\in\mathbb{C}$| and |$\mathcal{E}_{\mathbf{M}}$|s are given in Section 1.2, then the series

$$\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^l}^2=\sum_{\mathbf{M}}\left|a_{\mathbf{M}}\right|^2\lambda_{\mathbf{M}}^l$$

converges. Given any |$\alpha\in\mathbb{R}$|⁠, we may find |$l\in\mathbb{N}_0$| such that |$l\gt2\alpha$|⁠. The convergence of the above series shows that the series

$$\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^{2\alpha}}^2=\left\langle\mathscr{D}_{\mathsf{k}}^\alpha(f),\mathscr{D}_{\mathsf{k}}^\alpha(f)\right\rangle=\sum_{\mathbf{M}}\left|a_{\mathbf{M}}\right|^2\lambda_{\mathbf{M}}^{2\alpha}$$

converges, noting that |$\lambda_{\mathbf{M}}\in\mathbb{R}_{\gt1}$|⁠. By Cauchy–Schwarz,

$$\left\|f\right\|_{\mathscr{D}_{\mathsf{k}}^\alpha}^4=\left|\left\langle\mathscr{D}_{\mathsf{k}}^\alpha(f),f\right\rangle\right|^2{\leqslant}\left\langle\mathscr{D}_{\mathsf{k}}^\alpha(f),\mathscr{D}_{\mathsf{k}}^\alpha(f)\right\rangle\cdot\left\langle f,f\right\rangle\lt{\infty}.$$

Hence, |$f\in I(\sigma)_\alpha$|⁠.

Remark 5.6

G does not act on |$I(\sigma)_\alpha$|⁠.

Proof of Theorem 1.5

We will show that Z_s is a continuous functional on |$\big(J(\sigma),\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}\big)$|⁠. Then the conclusion is naturally extended to |$\big(I(\sigma)_1,\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}\big)$| since, be definition, |$J(\sigma)$| is a dense subspace of |$I(\sigma)_1$| with respect to |$\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}$|⁠.

By [13, Theorem 5.3], for tempered |$I(\sigma)$|⁠, the integrals |$Z_s(f)$| converge absolutely for any |$f\in I(\sigma)$| and |$\mathrm{Re}(s)\gt0$|⁠. As |$f\in J(\sigma)$| is a Schwartz function when restricted to N, the integral |$\Lambda_s(f)$| converges absolutely for any |$f\in J(\sigma)$| and |$\mathrm{Re}(s)\in(0,1)$|⁠. The argument in the proof of [15, Theorem 1.2], especially where the gamma factor arises, still works. So the relation (28) holds in our setup and, therefore, the continuity of Z_s on |$\big(J(\sigma),\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}\big)$| is equivalent to the continuity of |$\Lambda_s$| on |$\big(J(\sigma),\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}\big)$|⁠. Consider the maps

$$\big(J(\sigma),\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}\big)\xrightarrow{\quad \mathcal{T}_\psi\quad}\big(J(\sigma),\left\|\,\right\|_{\mathscr{D}_{\mathsf{k}}^1}\big)\xrightarrow{\quad\Lambda_s\quad}\mathbb{C}.$$

By Proposition 5.4, the continuity of |$\Lambda_s$| is equivalent to the continuity of |$\Lambda_s\circ\mathcal{T}_\psi$|⁠. Therefore, it suffices to show that the series

$$ \sum_{\mathbf{M}}\left|\Lambda_s\circ\mathcal{T}_\psi\left(\mathcal{E}_{\mathbf{M}}^{[1]}\right)\right|^2 $$

(5)

converges for |$\mathrm{Re}(s)\in(0,1)$|⁠, where |$\mathcal{E}_{\mathbf{M}}^{[1]}$|s form an orthonormal basis of |$\big(J(\sigma),\left\langle\,,\,\right\rangle_{\mathscr{D}_{\mathsf{k}}^1}\big)$|⁠. By definition,

$$\begin{aligned} &\ \quad\Lambda_s\circ\mathcal{T}_\psi\left(\mathcal{E}_{\mathbf{M}}^{[1]}\right)\notag\\ &=\int_{R\cap N}\int_{\mathsf{k}^\times}\mathcal{T}_\psi\left(\mathcal{E}_{\mathbf{M}}^{[1]}\right)\left(w_1u^{\prime}[y]\right)|y|_{\mathsf{k}}^{s-\frac{n-1}{2}}\psi^{-1}(u^{\prime})\mathrm{d}^\times y\mathrm{d} u^{\prime}\notag\\ &=\int_{R\cap N}\int_{\mathsf{k}^\times}\mathcal{T}_\psi\left(\mathcal{E}_{\mathbf{M}}^{[1]}\right)\left([y]\cdot[y^{-1}]w_1[y]\cdot[y^{-1}]u^{\prime}[y]\right)|y|_{\mathsf{k}}^{s-\frac{n-1}{2}}\psi^{-1}(u^{\prime})\mathrm{d}^\times y\mathrm{d} u^{\prime}\notag\\ &=\int_{\mathsf{k}^{(n^2-n)/2-1}}\int_{\mathsf{k}^\times}\sigma([y])\bar\rho([y])e^{[1]}_{\mathbf{m}_{1,2}}(y^{-1})\cdot\prod_{j=3}^n e^{[1]}_{\mathbf{m}_{1,j}}\big(\tfrac{u_{1,j}+u_{2,j}}{y}\big)\cdot\prod_{\substack{1{\leqslant} i\lt j{\leqslant} n\\ i\ne 1}}e_{\mathbf{m}_{i,j}}^{[1]}(u_{i,j})|y|_{\mathsf{k}}^{s-\frac{n-1}{2}}\mathrm{d}^\times y\prod_{\substack{1{\leqslant} i\lt j{\leqslant} n\\ j\ne2}}\mathrm{d} u_{i,j}\notag\\ &=\int_{\mathsf{k}^\times}\sigma([y^{-1}])\bar\rho([y^{-1}])e^{[1]}_{\mathbf{m}_{1,2}}(y)|y|_{\mathsf{k}}^{\frac{n-1}{2}-s-(n-2)}\mathrm{d}^\times y\cdot\prod_{\substack{1{\leqslant} i\lt j{\leqslant} n\\ j\ne2}}\int_{\mathsf{k}} e^{[1]}_{\mathbf{m}_{i,j}}(u_{i,j})\mathrm{d} u_{i,j},\label{L} \end{aligned}$$

where we have made the the change of variables |$u_{1,j}\mapsto y(u_{1,j}-u_{2,j})$| (for |$j{\geqslant}3$|⁠) and |$y\mapsto y^{-1}$| in the last step. Now we may write

$$ \sum_{\mathbf{M}}\left|\Lambda_s\circ\mathcal{T}_\psi\left(\mathcal{E}_{\mathbf{M}}^{[1]}\right)\right|^2=\prod_{1{\leqslant} i\lt j{\leqslant} n}\left(\sum_{\ \mathbf{m}_{i,j}}\big|S_{\mathbf{m}_{i,j}}\big|^2\right), $$

(5)

where |$\mathbf{m}_{i, j}$| ranges over |$\mathbb{N}_0$| (for |$\mathsf{k}=\mathbb{R}$|⁠) or |$\mathbb{N}_0\times\mathbb{N}_0$| (for |$\mathsf{k}=\mathbb{C}$|⁠) and

$$\begin{aligned}S_{\mathbf{m}_{i,j}}=\left\{\begin{array}{cl}\int_{\mathsf{k}^\times}\sigma([y^{-1}])\bar\rho([y^{-1}])e^{[1]}_{\mathbf{m}_{1,2}}(y)|y|_{\mathsf{k}}^{\frac{n-1}{2}-s-(n-2)}\mathrm{d}^\times y,&\text{for }(i,j)=(1,2),\\ \int_{\mathsf{k}} e^{[1]}_{\mathbf{m}_{i,j}}(x)\mathrm{d} x,&\text{for }(i,j)\ne(1,2).\end{array}\right.\end{aligned}$$

It remains to show that each series

$$ \sum_{\mathbf{m}_{i,j}}\left|S_{\mathbf{m}_{i,j}}\right|^2,\qquad1{\leqslant} i\lt j{\leqslant} n, $$

(5)

converges. We will deal with the series for the real and complex cases, respectively.

First, let |$\mathsf{k}=\mathbb{R}$|⁠. Write |$\mathbf{m}_{i, j}$| as |$m_{i, j}$|⁠. As |$I(\sigma)$| is assumed to be tempered, there exists |$s_i\in\mathbf{i}\,\mathbb{R}$| such that

$$\sigma_i=|\,|^{s_i}\quad\text{or}\quad\mathrm{sgn}\cdot|\,|^{s_i}.$$

If |$\sigma_1(x)=|x|_{\mathbb{R}}^{s_1}$|⁠, then
$$S_{m_{1,2}}=Z_{1-s_1-s,\mathrm{triv}}\big(e^{[1]}_{m_{1,2}}\big),$$
where |$Z_{s,\chi}$| denotes the local zeta integral in Section 1.1, and ‘|$\mathrm{triv}$|’ denotes the trivial character of |$\mathbb{R}^\times$|⁠. We may imitate the proof of Theorem 1.1 to show that the series |$\sum_{m_{1,2}\in\mathbb{N}_0}\big|S_{m_{1,2}}\big|^2$| converges for |$\mathrm{Re}(s)\in(0,1)$|⁠.
If |$\sigma_1(x)=\mathrm{sgn}(x)\cdot|x|_{\mathbb{R}}^{s_1}$|⁠, then
$$S_{m_{1,2}}=\begin{cases}0,&\text{if\, }m_{1,2}\in2\mathbb{N}_0,\\ 2\int_0^{\infty} e_{m_{1,2}}^{[1]}(y)|y|^{1-s_1-s}\frac{\mathrm{d} y}{|y|},&\text{if\, }m_{1,2}\in2\mathbb{N}_0+1.\end{cases}$$
In this case, we may use formula 7.376.3 of [9] to calculate |$S_{m_{1,2}}$| and then apply the same argument as the preceding case to show that the series |$\sum_{m_{1,2}\in\mathbb{N}_0}\big|S_{m_{1,2}}\big|^2$| also converges for |$\mathrm{Re}(s)\in(0,1)$|⁠.

For each |$(i, j)\ne(1,2)$|⁠, we have |$S_{m_{i, j}}=Z_{1,\mathrm{triv}}\big(e^{[1]}_{m_{i, j}}\big)$|⁠. By Theorem 1.1, the series |$\sum_{m_{i, j}\in\mathbb{N}_0}\big|S_{m_{i, j}}\big|^2$| converges. This proves Theorem 1.5 for the case |$\mathsf{k}=\mathbb{R}$|⁠.

Now let |$\mathsf{k}=\mathbb{C}$|⁠. Write |$\mathbf{m}_{i, j}=(m_{i, j},n_{i, j})\in\mathbb{N}_0\times\mathbb{N}_0$|⁠. As |$J(\sigma)$| is tempered, σ₁ takes the form

$$\sigma_1=\phi_\tau\cdot|\,|_{\mathbb{C}}^{s_1},$$

where |$\tau\in\mathbb{Z}$|⁠, |$s_1\in\mathbf{i}\,\mathbb{R}$| and |$\phi_\tau(z)=e^{\mathbf{i}\tau\theta}$| for |$z=re^{\mathbf{i}\theta}\in\mathbb{C}$|⁠. So we have

$$ \begin{aligned}S_{\mathbf{m}_{1,2}}&=\int_{\mathbb{C}^\times}e_{\mathbf{m}_{1,2}}^{[1]}(z)\phi_\tau\big(z^{-1}\big)(z\bar z)^{-(s_1+s)}\mathrm{d} z\\ &=\left\lbrack\pi\lambda_{\mathbf{m}_{1,2}}(m_{1,2})!(n_{1,2})!\right\rbrack^{-\frac12}\int_{\mathbb{C}^\times}H_{\mathbf{m}_{1,2}}(z,\bar z)e^{-\frac{z\bar z}{2}}\phi_\tau\big(z^{-1}\big)(z\bar z)^{-(s_1+s)}\mathrm{d} z. \end{aligned}$$

The explicit formula

$$H_{i,j}(z,\bar z)=i!j!\sum_{k=0}^{\min(i,\,j)}\frac{(-1)^k}{k!}\frac{z^{i-k}\bar z^{j-k}}{(i-k)!(j-k)!}$$

implies that

$$H_{i,j}(z,\bar z)=H_{i,j}(r,r)e^{(i-j)\mathbf{i}\theta},\qquad z=re^{\mathbf{i}\theta}.$$

Substituting it into |$S_{\mathbf{m}_{1,2}}$|⁠, we get that |$S_{\mathbf{m}_{1,2}}=0$| for |$m_{1,2}-n_{1,2}\ne\tau$|⁠. Below let us assume that |$m_{1,2}-n_{1,2}=\tau$|⁠. Under this assumption, the polar coordinate gives

$$ S_{\mathbf{m}_{1,2}}=2\pi\left\lbrack\pi\lambda_{\mathbf{m}_{1,2}}(m_{1,2})!(n_{1,2})!\right\rbrack^{-\frac12}\int_0^{\infty} H_{m_{1,2},n_{1,2}}(r,r)e^{-\frac{r^2}{2}}r^{1-2(s_1+s)}\mathrm{d} r. $$

(34)

Assume that |$\tau{\geqslant} 0$|⁠. Then |$H_{m_{1,2},n_{1,2}}(r, r)=(-1)^{n_{1,2}}(n_{1,2})!r^\tau L_{n_{1,2}}^{(\tau)}(r^2)$| by [6, formula 2.6.14], whence
$$ S_{\mathbf{m}_{1,2}}=(-1)^{n_{1,2}}2\left(\frac{\pi(n_{1,2})!}{\lambda_{\mathbf{m}_{1,2}}(n_{1,2}+\tau)!}\right)^{\frac12}\int_0^{\infty} L_{n_{1,2}}^{(\tau)}(r^2)e^{-\frac{r^2}{2}}r^{\tau+1-2(s_1+s)}\mathrm{d} r. $$
Making the change of variables |$r^2\mapsto x$| and applying [9, formula 7.414.7] yield
$$ S_{\mathbf{m}_{1,2}}=(-1)^{n_{1,2}}\left(\frac{\pi(n_{1,2}+\tau)!}{\lambda_{\mathbf{m}_{1,2}}(n_{1,2})!}\right)^{\frac12}\frac{\Gamma\big(1+\frac\tau2-(s_1+s)\big)}{\tau!}2^{1+\frac\tau2-(s_1+s)}F\big(-n_{1,2},1+\tfrac\tau2-(s_1+s);\tau+1;2\big). $$
Under our assumptions on τ, s and s₁, we have |$\tau+1\gt\mathrm{Re}\big(1+\frac\tau2-(s+s_1)\big)\gt0$|⁠. So we may use Euler’s integral formula for hypergeometric series to get
$$S_{\mathbf{m}_{1,2}}=(-1)^{n_{1,2}}\left(\frac{\pi(n_{1,2}+\tau)!}{\lambda_{\mathbf{m}_{1,2}}(n_{1,2})!}\right)^{\frac12}\frac{2^{1+\frac\tau2-(s_1+s)}}{\Gamma\big(\frac\tau2+s_1+s\big)}I_{n_{1,2}}\big(\tfrac\tau2-(s_1+s),\tfrac\tau2+(s_1+s)-1\big).$$
It follows that the convergence of |$\sum_{\mathbf{m}_{1,2}}|S_{\mathbf{m}_{1,2}}|^2$| amounts to the convergence of
$$ \sum_{n_{1,2}\in\mathbb{N}_0}\frac{(n_{1,2}+\tau)!}{\lambda_{\mathbf{m}_{1,2}}(n_{1,2})!}\left|I_{n_{1,2}}\big(\tfrac\tau2-(s_1+s),\tfrac\tau2+(s_1+s)-1\big)\right|^2. $$
By Lemma 3.1, for any ɛ > 0 and |$n_{1,2}\in\mathbb{N}$|⁠,
$$ \left|I_{n_{1,2}}\big(\tfrac\tau2-(s_1+s),\tfrac\tau2+(s_1+s)-1\big)\right|\ll_{_\varepsilon} n_{1,2}^{-\min\left\lbrace1+\frac\tau2-\mathrm{Re}(s),\,\frac\tau2+\mathrm{Re}(s)\right\rbrace+\varepsilon}. $$
Combining this estimate with the conditions |$\mathrm{Re}(s)\in(0,1)$|⁠, |$\lambda_{\mathbf{m}_{1,2}}=4n_{1,2}+2\tau+3$| and |$\frac{(n_{1,2}+\tau)!}{(n_{1,2})!}\sim n_{1,2}^\tau$| (as |$n_{1,2}$| tends to |${\infty}$|⁠) shows that the series (35) indeed converges.
Assume that τ < 0. As |$H_{i, j}(x, x)=H_{j, i}(x, x)$| for |$x\in\mathbb{R}$|⁠, by (34),
$$S_{\mathbf{m}_{1,2}}=2\pi\left\lbrack\pi\lambda_{\mathbf{m}_{1,2}}(m_{1,2})!(n_{1,2})!\right\rbrack^{-\frac12}\int_0^{\infty} H_{n_{1,2},m_{1,2}}(r,r)e^{-\frac{r^2}{2}}r^{1-2(s_1+s)}\mathrm{d} r,$$
which reduces the calculation of |$S_{\mathbf{m}_{1,2}}$| to the previous case. The series |$\sum_{\mathbf{m}_{1,2}}|S_{\mathbf{m}_{1,2}}|^2$| therefore also converges.

For each |$(i, j)\ne(1,2)$|⁠, since |$S_{\mathbf{m}_{i, j}}=Z_{1,\mathrm{triv}}\big(e_{\mathbf{m}_{i, j}}^{[1]}\big)$|⁠, the series |$\sum_{\mathbf{m}_{i, j}}\big|S_{\mathbf{m}_{i, j}}\big|^2$| converges by Theorem 1.1. This proves the conclusion for the complex case.

Remark 5.7

When defining |$\mathscr{D}_{\mathsf{k}}$|⁠, we put |$D_{\mathsf{k}}$| on each position of N. This is necessary: without |$D_{\mathsf{k}}$| acting on |$f|_{N_{i, j}}$|⁠, the series |$\sum_{\mathbf{m}_{i, j}}\big|S_{\mathbf{m}_{i, j}}\big|^2$| would be equal to |$\sum_{\mathbf{m}_{i, j}}|Z_{1,\mathrm{triv}}(e_{\mathbf{m}_{i, j}}^{[0]})|^2$| which diverges as Remark 3.4 (3) has pointed out, and hence the Rankin–Selberg integral would not be continuous with respect to the resulting Sobolev norm.

Theorem 5.8

Assume that |$I(\sigma)$| is tempered and |$\mathrm{Re}(s)\in(0,1)$|⁠. Then there exist constants C, |$C^{\prime}\gt0$| (depending on β in the complex case) such that the operator norm of |$\Lambda_s$| on |$(J(\sigma),\|\,\|_{\mathscr{D}_{\mathsf{k}}^1})$| satisfies

$$\left\|\Lambda_s\right\|_{\text{op}}{\geqslant} \left\lbrack C\cdot\left|s+s_1\right|^2+C^{\prime}\right\rbrack^{-\frac12}.$$

Proof.

Let ξ be a non-negative real-valued smooth function on |$\mathbb{R}$| with compact support lying in (1, 2). Define

$$\xi_{\mathsf{k}}=\xi\circ|\,|_{\mathsf{k}}\in\mathcal{S}(\mathsf{k}).$$

Set

$$\Phi_s=(\sigma\cdot\overline\rho)|_{\bar B}{\otimes}\left(\underset{1{\leqslant} i\l tj{\leqslant} n}{{\bigotimes}}\varphi_{i,j}\right)\in J(\sigma)$$

such that

|$\tau_{1,2}^\ast(\varphi_{1,2})=\sigma_1\cdot|\,|_{\mathsf{k}}^s\cdot\xi_{\mathsf{k}}$|⁠,
|$\tau_{i, i+1}^\ast(\varphi_{i, i+1})=T_\psi(\xi)$| for |$i=2,\cdots,n-1$|⁠,
|$\tau_{i, j}^\ast(\varphi_{i, j})=\xi$| for |$2{\leqslant} i+1\lt j{\leqslant} n$|⁠.

Then

$$ \Lambda_s\left(\Phi_s\right)=\int_{\mathsf{k}^\times}\xi_{\mathsf{k}}(x)\mathrm{d} x\cdot\left(\int_{\mathsf{k}}\xi(x)\mathrm{d} x\right)^{\frac{n(n-1)}{2}-1}, $$

which is a positive constant.

Next we calculate |$\left\|\Phi_s\right\|_{\mathscr{D}_{\mathsf{k}}^1}^2=\prod_{1{\leqslant} i \l tj{\leqslant} n}\left\langle\varphi_{i, j},\varphi_{i, j}\right\rangle_{D_{\mathsf{k}}^1}$| by imitating the argument/calculation in the proof of Theorem 1.3, with details omitted. The conclusion is: there exist positive constants C₀, C₁, C₂ depending on β when |$\mathsf{k}=\mathbb{C}$| (recall that |$\sigma_1=\phi_\beta\cdot|\,|_{\mathbb{C}}^{s_1}$| when |$\mathsf{k}=\mathbb{C}$|⁠) such that

|$\left\langle\varphi_{1,2},\varphi_{1,2}\right\rangle_{D_{\mathsf{k}}^1}{\leqslant} C_1\cdot|s+s_1|^2+C_2$|⁠,
|$\left\langle\varphi_{i, j},\varphi_{i, j}\right\rangle_{D_{\mathsf{k}}^1}{\leqslant} C_0$| for i < j.

As |$\left\|\Lambda_s\right\|_{\text{op}}{\geqslant}\frac{\left|\Lambda_s(\Phi_s)\right|}{\left\|\Phi_s\right\|_{\mathscr{D}_{\mathsf{k}}^1}}$|⁠, the theorem follows.

Remark 5.9

As Remark 4.1 has pointed out, inserting specific ξ in the proof gives explicit C.

Proof of Theorem 1.6

We adopt the notation ρ in Tate’s thesis [20, Section 2.4]. One should distinguish ρ from |$\bar\rho$| that occurs in the definition of |$I(\sigma)$|⁠. The gamma factor |$\gamma(s,\sigma_1,\psi)$| is generated in the proof of Theorem 1.4 of [15]:

$$\int_{\mathsf{k}^\times}f(x)\sigma_1(x)^{-1}|x|_{\mathsf{k}}^{1-s}\mathrm{d}^\times x=\gamma(s,\sigma_1,\psi)\int_{\mathsf{k}^\times}\mathcal{F}_{\psi^{-1}}(f)(x)\sigma_1(x)|x|_{\mathsf{k}}^s\mathrm{d}^\times x,\qquad f\in\mathcal{S}(\mathsf{k}),$$

where |$\mathcal{F}_{\psi^{-1}}$| is as defined in [15]. Note that the additive character ψ of |$\mathsf{k}$| in [20] is in special/standard form (see Section 2.2 therein), while ours is more general (see the definition above Lemma 5.1). Through the change of variables |$x\mapsto\frac{-\mu}{2\pi}x$|⁠, the preceding identity reads

$$\int_{\mathsf{k}^\times}f(x)c(x)\mathrm{d}^\times x=\gamma(s,\sigma_1,\psi)\sigma_1\big(\tfrac{-\mu}{2\pi}\big)^{-1}\left|\tfrac{\mu}{2\pi}\right|_{\mathsf{k}}^{-s}\int_{\mathsf{k}^\times}\skew7\widehat f(x)\skew5\widehat c(x)\mathrm{d}^\times x,$$

where |$\skew7\widehat {f}$| denotes the Fourier transform defined in [20, Theorem 2.2.2] and

$$c(x)=\sigma_1(x)^{-1}\cdot|x|_{\mathsf{k}}^{1-s},\qquad\skew5\widehat c(x)=|x|_{\mathsf{k}}\cdot c(x)^{-1},\qquad x\in\mathsf{k}^\times.$$

It follows from [20, Theorem 2.4.1] that

$$\gamma(s,\sigma_1,\psi)=\rho(c)\sigma_1\big(\tfrac{-\mu}{2\pi}\big)\left|\tfrac{\mu}{2\pi}\right|_{\mathsf{k}}^s.$$

As ρ was computed in [20, Section 2.5], we get the explicit formula for |$\gamma(s,\sigma_1,\psi)$|⁠. By Stirling’s asymptotic formula of Gamma function, we have for fixed |$\mathrm{Re}(s)\in(0,1)$| and |$s_1\in\mathbf{i}\,\mathbb{R}$| that

$$ \lim_{|\mathrm{Im}(s)|\to{\infty}}\left|\gamma(s,\sigma_1,\psi)\right|=(2\pi)^{-\frac{\theta_{\mathsf{k}}}{2}}|\mu|_{\mathsf{k}}^{\mathrm{Re}(s)}\big|\mathrm{Im}(s)\big|^{\theta_{\mathsf{k}}\cdot\left(\frac12-\mathrm{Re}(s)\right)}, $$

(36)

where |$\theta_{\mathsf{k}}$| is as in Section 1.1. Applying (36) and Theorem 5.8 to the identity (28), we obtain the theorem.

6. THE CASE OF|$\;{\mathrm{GL}}_2(\mathbb{R})$|

In this section, we will consider another type of differential operators and the associated Sobolev norms for the tempered principal series representations of |$G=\mathrm{GL}_2(\mathbb{R})$|⁠. Our aim is as before, namely, we will study the continuity and the lower bound of the operator norm of Rankin–Selberg integral with respect to the Sobolev norm.

Denote |$K=\mathrm{SO}_2(\mathbb{R})$|⁠. Then

$$ E= \begin{bmatrix} 0&1 \\- 1&0 \end{bmatrix} $$

forms a basis of the Lie algebra of K. When |$I(\sigma)$| is tempered, picking up those elements in |$I(\sigma)$| with finite (standard) L²-norm and taking the completion of the resulting subspace give a unitary representation of G under the right regular translation, denoted |$I(\sigma)^\flat$|⁠. Set

$$u_x=\begin{bmatrix}1&x\\0&1\end{bmatrix}\in N,\qquad x\in\mathbb{R}.$$

Fix an orthonormal basis |$\left\lbrace\frac{1}{\sqrt\pi}e^{2n\mathbf{i} x}\right\rbrace_{n\in\mathbb{Z}}$| of |$L^2\big(\mathbb{R}/\mathbb{Z}\pi\big)$|⁠. Then we have the K-type vectors of |$I(\sigma)^\flat$|⁠:

$$\begin{aligned}\mathfrak{f}_n(\bar bu_x)=(\sigma{\otimes}\bar\rho)(\bar b)\cdot \frac{1}{\sqrt\pi}e^{2n\mathbf{i}\arctan x}(1+x^2)^{\frac{s_1-s_2-1}{2}},\qquad n\in\mathbb{Z},\quad \bar b\in\bar B,\end{aligned}$$

which form an orthonormal basis of |$I(\sigma)^\flat$| with respect to the inner product |$\left\langle\,,\,\right\rangle$| as in (1).

Lemma 6.1

|$E(\mathfrak{f}_n)=2n\mathbf{i}\cdot\mathfrak{f}_n$| for any |$n\in\mathbb{Z}$|⁠.

Proof.

By definition,

$$E(f|_N)(u_x)=\frac{\mathrm{d}}{\mathrm{d} t}\left(\eta_\sigma\big(e^{tE}\big)f(u_x)\right)\Big|_{t=0}=(1-s_1+s_2)xf(u_x)+(1+x^2)\frac{\mathrm{d} f}{\mathrm{d} x}(u_x),\qquad f\in I(\sigma),$$

where η_σ is as in Section 1.2. The rest follows from direct computation.

Given any δ > 0, set

$$\mathcal{D}_\delta=-E^2+\delta.$$

Corollary 6.2

|$\mathcal{D}_\delta$| is a self-adjoint positive operator on |$\big(I(\sigma)^\flat,\left\langle\,,\,\right\rangle\big)$|⁠.

Proof.

The adjoint of E is −E. So |$-E^2=(-E)E$| is self-adjoint. The positivity of |$\mathcal{D}_\delta$| follows from the preceding lemma: |$\mathcal{D}_\delta(\mathfrak{f}_n)=(4n^2+\delta)\mathfrak{f}_n$|⁠.

For any |$\alpha\in\mathbb{R}$|⁠, define the operator |$\mathcal{D}_\delta^\alpha$| on |$I(\sigma)^\flat$|⁠:

$$\mathcal{D}_\delta^\alpha(\mathfrak{f}_n)=(4n^2+\delta)^\alpha\cdot\mathfrak{f}_n,\qquad n\in\mathbb{Z}.$$

Then |$\mathcal{D}_\delta^\alpha$| is self-adjoint and positive on |$I(\sigma)^\flat$|⁠. We have the associated inner product

$$\left\langle f,g\right\rangle_{\mathcal{D}_\delta^\alpha}=\left\langle\mathcal{D}_\delta^\alpha(f),g\right\rangle,\qquad f,\ g\in I(\sigma)^\flat,$$

where |$\left\langle\,,\,\right\rangle$| is as in (1). Denote

$$\left\|f\right\|_{\mathcal{D}_\delta^\alpha}=\left\langle f,f\right\rangle_{\mathcal{D}_\delta^\alpha}^{\frac12},\qquad f\in I(\sigma)^\flat.$$

The vectors

$$\mathfrak{b}_n^{[\alpha]}=(4n^2+\delta)^{-\frac\alpha2}\cdot\mathfrak{f}_n,\qquad n\in\mathbb{Z},$$

form an orthonormal basis of

$$\mathbb{I}(\sigma)_\alpha=\left\lbrace f\in I(\sigma):\left\|f\right\|_{\mathcal{D}_\delta^\alpha}\lt{\infty}\right\rbrace,$$

with respect to |$\left\langle\,,\,\right\rangle_{\mathcal{D}_\delta^\alpha}$|⁠. Using the same idea with Lemma 5.5, we may prove the following lemma.

Lemma 6.3

|$\mathbb{I}(\sigma)_\alpha$| is a subspace of |$I(\sigma)$| containing |$J(\sigma)$|⁠.

Theorem 6.4

Assume that |$I(\sigma)$| is tempered. For any |$s\in\mathbb{C}$| with |$\mathrm{Re}(s)\in(0,1)$| and any |$\alpha\gt|\frac12-\mathrm{Re}(s)|$|⁠, the Rankin–Selberg integral Z_s is a continuous operator on |$\mathbb{I}(\sigma)_\alpha$| with respect to the Sobolev norm |$\left\|\,\right\|_{\mathcal{D}_\delta^\alpha}$|⁠.

Proof.

For tempered |$I(\sigma)$| and |$\mathrm{Re}(s)\in(0,1)$|⁠, both integrals |$Z_s(f)$| and |$\Lambda_s(f)$| converge absolutely for all |$f\in I(\sigma)$| (note that the first condition in (26) is void for the present case) and the funtionals Z_s and |$\Lambda_s$| both lie in |$\mathrm{Hom}_R(I(\sigma),\psi_s)$| which is at most one-dimensional by the uniqueness of Rankin–Selberg integrals (see [5]), whence Z_s differ from |$\Lambda_s$| by a constant scalar. Inserting |$f\in J(\sigma)$| into Z_s gives the scalar which is precisely the gamma factor given in (28). As a consequence, the continuity of Z_s on |$\big(\mathbb{I}(\sigma)_\alpha,\left\|\,\right\|_{\mathcal{D}_\delta^\alpha}\big)$| amounts to the continuity of |$\Lambda_s$| on |$\big(\mathbb{I}(\sigma)_\alpha,\left\|\,\right\|_{\mathcal{D}_\delta^\alpha}\big)$|⁠, and we shall show that the series

$$ \sum_{n\in\mathbb{Z}}\left|\Lambda_s\left(\mathfrak{b}_n^{[\alpha]}\right)\right|^2 $$

(37)

converges under the prescribed conditions. By definition, we have for |$f\in I(\sigma)$| that

$$ \begin{aligned}\Lambda_s\left(f\right)&=\int_{\mathbb{R}^\times}f\left(w_1[x]\right)|x|^{s-\frac12}\mathrm{d}^\times x\notag\\ &=\int_{\mathbb{R}^\times}f\left([x]u_{x^{-1}}\right)|x|^{s-\frac12}\mathrm{d}^\times x\notag\\ &=\int_{\mathbb{R}^\times}f\left(u_{x^{-1}}\right)|x|^{s+s_1-1}\mathrm{d}^\times x \end{aligned}$$

(38)

$$\begin{aligned} &=\int_{\mathbb{R}^\times}f\left(u_x\right)|x|^{1-s-s_1}\mathrm{d}^\times x, \end{aligned}$$

(39)

where (38) follows from the definition of |$I(\sigma)$| and (39) follows from the change of variables |$x\mapsto x^{-1}$|⁠. Inserting |$f=\mathfrak{b}_n^{[\alpha]}$| and making the change of variables |$x\mapsto\tan y$| yield

$$\begin{aligned} \Lambda_s\left(\mathfrak{b}_n^{[\alpha]}\right)=\pi^{-\frac12}(4n^2+\delta)^{-\frac\delta2}\int_{-\frac\pi2}^{\frac\pi2}e^{2n\mathbf{i} y}(\cos y)^{-1+s_2+s}|\sin y|^{-s_1-s}\mathrm{d} y.\label{leq-3} \end{aligned}$$

Below we deal with the integral

$$\int_{-\frac\pi2}^{\frac\pi2}e^{2n\mathbf{i} y}(\cos y)^{-1+s_2+s}|\sin y|^{-s_1-s}\mathrm{d} y=\mathcal{J}_n+\mathcal{J}_n^{\prime},$$

where

$$ \mathcal{J}_n=\int_0^{\frac\pi2}e^{2n\mathbf{i} y}(\cos y)^{-1+s_2+s}|\sin y|^{-s_1-s}\mathrm{d} y,\qquad \mathcal{J}_n^{\prime}=\int_{-\frac\pi2}^0e^{2n\mathbf{i} y}(\cos y)^{-1+s_2+s}|\sin y|^{-s_1-s}\mathrm{d} y. $$

The change of variables |$y\mapsto -y$| shows that |$\mathcal{J}_n^{\prime}=\mathcal{J}_{-n}$|⁠. So it suffices to treat |$\mathcal{J}_n$|⁠. For |$\mathrm{Re}(s)\in(0,1)$|⁠, applying [9, formula 3.892.3] gives

$$ \begin{aligned}\mathcal{J}_n&=2^{s_1-s_2}e^{\mathbf{i}\pi\frac{2n-s_2-s}{2}}B\left(\frac{2n+1+s_1-s_2}{2},s_2+s\right)F\left(s_1+s,\frac{2n+1+s_1-s_2}{2};\frac{2n+1+s_1+s_2+2s}{2};-1\right)\\ &+2^{s_1-s_2}e^{\mathbf{i}\pi\frac{1-s_1-s}{2}}B\left(\frac{2n+1+s_1-s_2}{2},1-s_1-s\right)F\left(1-s_2-s,\frac{2n+1+s_1-s_2}{2};\frac{2n+3-s_1-s_2-2s}{2};-1\right). \end{aligned}$$

Here and henceforth, the conditions |$\mathrm{Re}(s)\in(0,1)$| and s₁, |$s_2\in\mathbf{i}\mathbb{R}$| are used in a crucial way.

Firstly, let us assume that |$n{\geqslant} 0$|⁠. In this case we use the classical formula |$B(z_1,z_2)=\frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1+z_2)}$| and Euler’s integral representation of hypergeometric series to get
$$\begin{aligned} \mathcal{J}_n&=2^{s_1-s_2}e^{\mathbf{i}\pi\frac{2n-s_2-s}{2}}\int_0^1 x^{\frac{2n-1+s_1-s_2}{2}}(1-x)^{s_2+s-1}(1+x)^{-s_1-s}\mathrm{d} x\\ &+2^{s_1-s_2}e^{\mathbf{i}\pi\frac{1-s_1-s}{2}}\int_0^1 x^{\frac{2n-1+s_1-s_2}{2}}(1-x)^{-s_1-s}(1+x)^{-1+s_2+s}\mathrm{d} x, \end{aligned}$$
which implies that
$$\begin{aligned} |\mathcal{J}_n|&{\leqslant} C_1\int_0^1 x^{\frac{2n-1}{2}}(1-x)^{\mathrm{Re}(s)-1}(1+x)^{-\mathrm{Re}(s)}\mathrm{d} x+C_2\int_0^1 x^{\frac{2n-1}{2}}(1-x)^{-\mathrm{Re}(s)}(1+x)^{\mathrm{Re}(s)-1}\mathrm{d} x\\ &{\leqslant} C_1^{\prime}\int_0^1 x^{\frac{2n-1}{2}}(1-x)^{\mathrm{Re}(s)-1}\mathrm{d} x+C_2^{\prime}\int_0^1 x^{\frac{2n-1}{2}}(1-x)^{-\mathrm{Re}(s)}\mathrm{d} x\\ &=C_1^{\prime}B\left(n+\tfrac12,\mathrm{Re}(s)\right)+C_2^{\prime}B\left(n+\tfrac12,1-\mathrm{Re}(s)\right), \end{aligned}$$
where C₁, C₂, |$C_1^{\prime}$| and |$C_2^{\prime}$| are positive constants depending on s₁, s₂ and s, and we have used the fact that both |$(1+x)^{\mathrm{Re}(s)-1}{\leqslant} 1$| and |$(1+x)^{-\mathrm{Re}(s)}{\leqslant} 1$| for any |$x\in[0,1]$|⁠. Using Stirling’s asymptotic formula for Gamma function, we have for |$x\in(0,1)$| that
$$B(n+\tfrac12,x)\,\sim\,\Gamma(x)n^{-x},\qquad \text{as}\ \,n\to+{\infty}.$$
So we get
$$|\mathcal{J}_n|\ll_{_{s_1,s_2,s}}n^{-\min\{\mathrm{Re}(s),\,1-\mathrm{Re}(s)\}}.$$
Secondly, we assume that n < 0. In this case,
$$\mathcal{J}_n=\overline{\int_0^{\frac{\pi}{2}}e^{2\mathbf{i}(-n)y}(\cos y)^{-1+\bar s_2+\bar s}(\sin y)^{-\bar s_1-\bar s}\mathrm{d} y},$$
where the integral under the conjugate bar can be treated as in the previous case (since |$-n\gt0$|⁠). The conclusion is
$$|\mathcal{J}_n|\ll_{_{s_1,s_2,s}}|n|^{-\min\{\mathrm{Re}(s),\,1-\mathrm{Re}(s)\}}.$$

Consequently,

$$\left|\Lambda_s\left(\mathfrak{b}_n^{[\alpha]}\right)\right|\ll_{_{s_1,s_2,s}}\left(4n^2+\delta\right)^{-\frac\alpha2}|n|^{-\min\{\mathrm{Re}(s),\,1-\mathrm{Re}(s)\}}.$$

Now it is clear that the series (37) converges for

$$\alpha\gt\tfrac12-\min\left\lbrace\mathrm{Re}(s),1-\mathrm{Re}(s)\right\rbrace=|\tfrac12-\mathrm{Re}(s)|.$$

Remark 6.5

Let |$s_1=s_2\in\mathbf{i}\mathbb{R}$| and |$s=-s_1+\varepsilon$|⁠, where ɛ is any fixed number in (0, 1). Then we have for |$n{\geqslant} 0$| that

$$\begin{aligned} \mathcal{J}_n&=(-1)^ne^{-\frac{\varepsilon\mathbf{i}\pi}{2}}\int_0^1x^{n-\frac12}(1-x)^{\varepsilon-1}(1+x)^{-\varepsilon}\mathrm{d} x+\mathbf{i} e^{-\frac{\varepsilon\mathbf{i}\pi}{2}}\int_0^1x^{n-\frac12}(1-x)^{-\varepsilon}(1+x)^{\varepsilon-1}\mathrm{d} x. \end{aligned}$$

As |$(1+x)^{-\varepsilon}\in[2^{-\varepsilon},1]$| for |$x\in[0,1]$|⁠, the integrals

$$\int_0^1x^{n-\frac12}(1-x)^{\varepsilon-1}(1+x)^{-\varepsilon}\mathrm{d} x\quad\text{and}\quad\int_0^1x^{n-\frac12}(1-x)^{\varepsilon-1}\mathrm{d} x$$

are bounded by each other, where the latter integral equals |$B(n+\frac12,\varepsilon)$| and has the order n^−ɛ as n tends to |$+{\infty}$|⁠. So the integral

$$\int_0^1x^{n-\frac12}(1-x)^{\varepsilon-1}(1+x)^{-\varepsilon}\mathrm{d} x$$

has order n^−ɛ, as n tends to |$+{\infty}$|⁠. For the similar reason, the integral

$$\int_0^1x^{n-\frac12}(1-x)^{-\varepsilon}(1+x)^{\varepsilon-1}\mathrm{d} x$$

has order |$n^{\varepsilon-1}$|⁠, as n tends to |$+{\infty}$|⁠. When |$\varepsilon\ne\frac12$|⁠, we have |$-\varepsilon\ne\varepsilon-1$| and whence |$|\mathcal{J}_n|$| has order |$|n|^{\max\left\lbrace-\varepsilon,\,\varepsilon-1\right\rbrace}$|⁠, as |$|n|\to{\infty}$|⁠. When |$\varepsilon=\frac12$|⁠,

$$\mathcal{J}_n=\begin{cases}\sqrt{2}\int_0^1x^{n-\frac12}(1-x)^{-\frac12}(1+x)^{-\frac12}\mathrm{d} x,&\text{if}\ \,n\in2\mathbb{N}_0+1,\\ \sqrt{2}\,\mathbf{i}\int_0^1x^{n-\frac12}(1-x)^{-\frac12}(1+x)^{-\frac12}\mathrm{d} x,&\text{if}\ \,n\in2\mathbb{N}_0.\end{cases}.$$

Repeating the argument in the proof shows that |$|\mathcal{J}_n|$| has order |$|n|^{-\frac12}$|⁠, as |$|n|$| tends to |${\infty}$|⁠. Then we may immediately verify that the series (37) fails to converge for α = 0. In other words, for s₁, s₂ and s satisfying the conditions in Theorem 6.4 plus the special conditions as in this remark, the Rankin–Selberg integral is not continuous with respect to the standard L²-norm of |$J(\sigma)$|⁠.

Theorem 6.6

Assume that |$I(\sigma)$| is tempered. For α = 1 and |$\mathrm{Re}(s)\in(0,1)$|⁠, the operator norm of the Rankin–Selberg integral on |$\left(\mathbb{I}(\sigma)_\alpha,\left\|\,\right\|_{\mathcal{D}_\delta^\alpha}\right)$| satisfies

$$\left\|Z_s\right\|_{\text{op}}{\geqslant} C\cdot|s|^{-\frac32+\mathrm{Re}(s)},\qquad\text{as}\ \,|s|\to{\infty},$$

where C is a positive constant depending on σ.

Proof.

Let f be a smooth real-valued non-negative function on |$\mathbb{R}$| whose support is nonempty and lies in (1, 2). Define |$f_s\in J(\sigma)$| so that

$$f_s|_N(u_x)=|x|^{s_1+s}f(x).$$

Then, by (39),

$$\Lambda_s(f_s)=\int_{\mathbb{R}^\times}f(x)\mathrm{d} x,$$

a positive constant depending on f. By definition, |$\left\langle f_s,f_s\right\rangle_{\mathcal{D}_\delta^1}=\delta\left\langle f_s,f_s\right\rangle+\left\langle E(f_s),E(f_s)\right\rangle$|⁠, where

$$\delta\left\langle f_s,f_s\right\rangle=\delta\int_{\mathbb{R}}|x|^{2\mathrm{Re}(s)}\left|f(x)\right|\mathrm{d} x$$

and

$$\begin{aligned} &\hphantom{=}\left\langle E(f_s),E(f_s)\right\rangle\\ &=\int_{\mathbb{R}}\left|(1-s_1+s_2)|x|^{1+s_1+s}f(x)+(s_1+s)(1+x^2)|x|^{s_1+s-1}f(x)+(1+x^2)|x|^{s_1+s}f^{\prime}(x)\right|^2\mathrm{d} x\\ &{\leqslant}3\int_{\mathbb{R}}\left|(1-s_1+s_2)x|x|^{s_1+s}f(x)\right|^2\mathrm{d} x+3\int_{\mathbb{R}}\left|(s_1+s)(1+x^2)|x|^{s_1+s-1}f(x)\right|^2\mathrm{d} x+3\int_{\mathbb{R}}\left|(1+x^2)|x|^{s_1+s}f^{\prime}(x)\right|^2\mathrm{d} x\\ &=3|1-s_1+s_2|^2\int_{\mathbb{R}}|x|^{2+2\mathrm{Re}(s)}|f(x)|^2\mathrm{d} x+3|s_1+s|^2\int_{\mathbb{R}}(1+x^2)^2|x|^{2\mathrm{Re}(s)-2}|f(x)|^2\mathrm{d} x+3\int_{\mathbb{R}}(1+x^2)^2|x|^{2\mathrm{Re}(s)}| \\ &f^{\prime}(x)|^2\mathrm{d} x =C_1\cdot|s_1+s_2|^2+C_2, \end{aligned}$$

where C₁ and C₂ are positive constants (C₂ depends on s₁, s₂). Now we have

$${\left\|\Lambda_s\right\|}_\text{op}\geqslant\frac{\left|\Lambda_s(f_s)\right|}{{\left\|f_s\right\|}_{\mathcal D_\delta^1}}\geqslant C_3\cdot\vert s\vert^{-1},\qquad\text{as}\;\,\vert s\vert\rightarrow\infty,$$

where C₃ depends on s₁ and s₂. The theorem then follows, like Theorem 1.6, from the relation (28) and the estimate (36).

Remark 6.7

Theorems 6.4 and 6.6 remain true if we replace |$\mathbb{I}(\sigma)_\alpha$| with |$J(\sigma)$|⁠.

Remark 6.8

Put

$$H=\begin{bmatrix}1&0\\0&-1\end{bmatrix},\qquad F=\begin{bmatrix}0&1\\1&0\end{bmatrix}.$$

Then H, E and F form a basis of |$\mathfrak{s}\mathfrak{l}_2(\mathbb{R})$|⁠. The actions of H and F on |$I(\sigma)$| are given by

$$\begin{aligned} H(f|_N)(u_x)&=\frac{\mathrm{d}}{\mathrm{d} t}\left(\eta_\sigma\big(e^{tH}\big)f(u_x)\right)\Big|_{t=0}=(s_1-s_2+1)f(u_x)-2x\frac{\mathrm{d} f}{\mathrm{d} x}(u_x),\\ F(f|_N)(u_x)&=\frac{\mathrm{d}}{\mathrm{d} t}\left(\eta_\sigma\big(e^{tF}\big)f(u_x)\right)\Big|_{t=0}=(s_1-s_2-1)xf(u_x)+(1-x^2)\frac{\mathrm{d} f}{\mathrm{d} x}(u_x), x\in\mathbb{R},\quad f\in I(\sigma). \end{aligned}$$

The adjoint of H, E and F are −H, −E, −F, respectively. Let |$\mathcal{U}(\mathfrak{g})$| be the universal enveloping algebra of |$\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2(\mathbb{R}){\otimes}\mathbb{C}$|⁠. Fix a, b, |$c\in\mathbb{C}$| such that

$$\mathcal{L}:=(aH+bE+cF)^\ast(aH+bE+cF)\in\mathcal{U}(\mathfrak{g})$$

commutes with E. Then |$\mathcal{L}$| is a self-adjoint second-order differential operator on |$\mathbb{I}(\sigma)_\alpha$|⁠, with coefficients in the polynomial ring |$\mathbb{R}[s_1,s_2,x]$|⁠. In addition, |$\mathfrak{f}_n$|s are eigenvectors of |$\mathcal{L}$|⁠, with non-negative eigenvalues taking the form |$f_2(s_1,s_2)n^2+f_1(s_1,s_2)n+f_0(s_1,s_2)$|⁠, where |$f_i(s_1,s_2)$|s are complex polynomials in s₁ and s₂. These eigenvalues tend to |${\infty}$|⁠, as n tends to |${\infty}$|⁠. Given any δ > 0, set |$\mathcal{L}_\delta=\mathcal{L}+\delta$|⁠. Then |$\mathcal{L}_\delta$| is positive and we may define |$\mathcal{L}_\delta^\alpha$|⁠, as well as the associated Sobolev norm |$\left\|\,\right\|_{\mathcal{L}_\delta^\alpha}$| for any |$\alpha\in\mathbb{R}$|⁠. Like before, we may study the continuity of Z_s and the lower bound of the operator norm of Z_s, with respect to |$\left\|\,\right\|_{\mathcal{L}_\delta^\alpha}$|⁠. In this respect, Theorems 6.4 and 6.6 are valid if we replace |$\mathcal{D}_\delta^\alpha$| with |$\mathcal{L}_\delta^\alpha$|⁠.

Acknowledgements

The authors are indebted to Binyong Sun for introducing them the theme of the paper and for his kind help. They would like to express their sincere gratitude to the anonymous referee for carefully reading the paper and making very useful suggestions.

Funding

F.S. was supported in part by the National Science Foundation of China (No. 12471007).

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