Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein $g_{k}$-functions and the Laplace type multipliers

TL;DR

This paper extends harmonic analysis related to Jacobi expansions by studying Littlewood-Paley-Stein $g_k$-functions and Laplace multipliers, establishing norm equivalences and multiplier results.

Contribution

It introduces and analyzes Littlewood-Paley-Stein $g_k$-functions for Jacobi operators, proving norm equivalences with weights and deriving Laplace multiplier results.

Findings

Established norm equivalence for $g_k^{(eta,eta)}$-functions with weights.

Proved boundedness of Laplace type multipliers on Jacobi expansions.

Extended harmonic analysis framework for Jacobi operators.

Abstract

The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}-I$, where $J^{(\alpha,\beta)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we define the corresponding Littlewood-Paley-Stein $g_k^{(\alpha,\beta)}$-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

Equations

$$a_n^{(\alpha ,\beta )}=\frac{2}{2n+\alpha +\beta +2}\sqrt{\frac{(n+1)(n+\alpha +1)(n+\beta +1)(n+\alpha +\beta +1)}{(2n+\alpha +\beta +1)(2n+\alpha +\beta +3)}},n\ge 1,$$

$$a_0^{(\alpha ,\beta )}=\frac{2}{\alpha +\beta +2}\sqrt{\frac{(\alpha +1)(\beta +1)}{(\alpha +\beta +3)}},$$

$$b_n^{(\alpha ,\beta )}=\frac{{\beta }^2-{\alpha }^2}{(2n+\alpha +\beta )(2n+\alpha +\beta +2)},n\ge 1,$$

$$b_0^{(\alpha ,\beta )}=\frac{\beta -\alpha }{\alpha +\beta +2}.$$

$$J^{(\alpha ,\beta )}f(n)=a_{n-1}^{(\alpha ,\beta )}f(n-1)+b_n^{(\alpha ,\beta )}f(n)+a_n^{(\alpha ,\beta )}f(n+1),n\ge 1,$$

$$P_n^{(\alpha ,\beta )}(x)=\frac{(-1{)}^n}{2^{n}n!}(1-x{)}^{-\alpha }(1+x{)}^{-\beta }\frac{d^n}{d{x}^n}\left\{(1-x{)}^{\alpha +n}(1+x{)}^{\beta +n}\right\},$$

$$d{\mu }_{\alpha ,\beta }(x)=(1-x{)}^{\alpha }(1+x{)}^{\beta }dx.$$

$$w_n^{(\alpha ,\beta )}$$

$$=\sqrt{\frac{(2n+\alpha +\beta +1)n!\Gamma (n+\alpha +\beta +1)}{2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}},n\ge 1,$$

$$w_0^{(\alpha ,\beta )}=\frac{1}{\Vert P_0^{(\alpha ,\beta )}{\Vert }_{L^2((-1,1),d{\mu }_{\alpha ,\beta })}}=\sqrt{\frac{\Gamma (\alpha +\beta +2)}{2^{\alpha +\beta +1}\Gamma (\alpha +1)\Gamma (\beta +1)}},$$

$$J^{(\alpha ,\beta )}{p}_n^{(\alpha ,\beta )}(x)=x{p}_n^{(\alpha ,\beta )}(x).$$

$${\mathcal{J}}^{(\alpha ,\beta )}f(n)=(J^{(\alpha ,\beta )}-I)f(n),x\in [-1,1],$$

$$W_t^{(\alpha ,\beta )}f(n)=\sum _{m=0}^{\infty }f(m)K_t^{(\alpha ,\beta )}(m,n),$$

$$K_t^{(\alpha ,\beta )}(m,n)={\int }_{-1}^1{e}^{-(1-x)t}{p}_m^{(\alpha ,\beta )}(x)p_n^{(\alpha ,\beta )}(x)d{\mu }_{\alpha ,\beta }(x).$$

$$g_k^{(\alpha ,\beta )}(f)(n)={\left({\int }_0^{\infty }{t}^{2k-1}{\left|\frac{{\partial }^k}{\partial t^k}{W}_t^{(\alpha ,\beta )}f(n)\right|}^{2}dt\right)}^{1/2},k\ge 1.$$

$${\Delta }_{d}f(n)=f(n-1)-2f(n)+f(n+1)$$

$$\ell^{p}(\mathbb{N},w)=\left\{f=\{f(n)\}_{n\geq 0}:\|f\|_{\ell^{p}(\mathbb{N},w)}:=\Bigg{(}\sum_{m=0}^{\infty}|f(m)|^{p}w(m)\Bigg{)}^{1/p}<\infty\right\},$$

$$\sup_{\begin{smallmatrix}0\leq n\leq m\\ n,m\in\mathbb{N}\end{smallmatrix}}\frac{1}{(m-n+1)^{p}}\Bigg{(}\sum_{k=n}^{m}w(k)\Bigg{)}\Bigg{(}\sum_{k=n}^{m}w(k)^{-1/(p-1)}\Bigg{)}^{p-1}<\infty,$$

$$C_1\Vert f{\Vert }_{{\ell }^p(\mathbb{N},w)}\le \Vert g_k^{(\alpha ,\beta )}(f){\Vert }_{{\ell }^p(\mathbb{N},w)}\le C_2\Vert f{\Vert }_{{\ell }^p(\mathbb{N},w)},f\in {\ell }^2(\mathbb{N})\cap {\ell }^p(\mathbb{N},w),$$

$$P_t^{(\alpha ,\beta )}f(n)=\frac{1}{\sqrt{\pi }}{\int }_0^{\infty }\frac{e^{-u}}{\sqrt{u}}{W}_{t^2/(4u)}^{(\alpha ,\beta )}f(n)du,t\ge 0,$$

$${\mathfrak{g}}_k^{(\alpha ,\beta )}(f)(n)={\left({\int }_0^{\infty }{t}^{2k-1}{\left|\frac{{\partial }^k}{\partial t^k}{P}_t^{(\alpha ,\beta )}f(n)\right|}^{2}dt\right)}^{1/2},k\ge 1.$$

$$C_1\Vert f{\Vert }_{{\ell }^p(\mathbb{N},w)}\le \Vert {\mathfrak{g}}_k^{(\alpha ,\beta )}(f){\Vert }_{{\ell }^p(\mathbb{N},w)}\le C_2\Vert f{\Vert }_{{\ell }^p(\mathbb{N},w)},f\in {\ell }^2(\mathbb{N})\cap {\ell }^p(\mathbb{N},w),$$

$$c_m^{(\alpha ,\beta )}(f)={\int }_{-1}^{1}f(x)p_m^{(\alpha ,\beta )}(x)d{\mu }_{\alpha ,\beta }(x)$$

$$f(x)=\sum _{m=0}^{\infty }{c}_m^{(\alpha ,\beta )}(f)p_m^{(\alpha ,\beta )}(x),$$

$$F_{\alpha ,\beta }(x)=\sum _{m=0}^{\infty }f(m)p_m^{(\alpha ,\beta )}(x)$$

$$\Vert f{\Vert }_{{\ell }^2(\mathbb{N})}=\Vert F_{\alpha ,\beta }{\Vert }_{L^2([-1,1],d{\mu }_{\alpha ,\beta })}$$

$$\sum _{m=0}^{\infty }f(m)g(m)={\int }_{-1}^1{F}_{\alpha ,\beta }(x)G_{\alpha ,\beta }(x)d{\mu }_{\alpha ,\beta }(x),f,g\in {\ell }^2(\mathbb{N}),$$

$$T_{M}f(n)=c_n^{(\alpha ,\beta )}(M(1-\cdot )F_{\alpha ,\beta }).$$

$$M(x)=x{\int }_0^{\infty }{e}^{-xt}a(t)dt,$$

$$\Vert T_{M}f{\Vert }_{{\ell }^p(\mathbb{N},w)}\le C\Vert f{\Vert }_{{\ell }^p(\mathbb{N},w)},f\in {\ell }^2(\mathbb{N})\cap {\ell }^p(\mathbb{N},w),$$

$$x^{i\gamma }=\frac{x}{\Gamma (1-i\gamma )}{\int }_0^{\infty }{e}^{-xt}{t}^{-i\gamma },\gamma \in \mathbb{R},$$

Full text

The first-named author was supported by a predoctoral research grant of the Government of Comunidad Autónoma de La Rioja. The second-named author was supported by grant MTM2015-65888-C04-4-P MINECO/FEDER, UE, from Spanish Government. The third-named author was supported by a predoctoral research grant of the University of La Rioja.

1. Introduction

We begin by setting some aspects of our context as in the previous papers [1, 3]. For $\alpha,\beta>-1$, we take the sequences $\{a_{n}^{(\alpha,\beta)}\}_{n\in\mathbb{N}}$ and $\{b_{n}^{(\alpha,\beta)}\}_{n\in\mathbb{N}}$ given by

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

For each sequence $\{f(n)\}_{n\geq 0}$, we define the operator $\{J^{(\alpha,\beta)}f(n)\}_{n\geq 0}$ by the relation

[TABLE]

and $J^{(\alpha,\beta)}f(0)=b_{0}^{(\alpha,\beta)}f(0)+a_{0}^{(\alpha,\beta)}f(1)$.

Defining the Jacobi polynomials $\{P^{(\alpha,\beta)}_{n}(x)\}_{n\geq 0}$ through the Rodrigues’ formula (see [20, p. 67, eq. (4.3.1)])

[TABLE]

it is well known that they are orthogonal on the interval $[-1,1]$ with respect to the measure

[TABLE]

Moreover, the sequence $\{p_{n}^{(\alpha,\beta)}(x)\}_{n\geq 0}$, given by $p_{n}^{(\alpha,\beta)}(x)=w_{n}^{(\alpha,\beta)}P_{n}^{(\alpha,\beta)}(x)$ where

[TABLE]

and

[TABLE]

is an orthonormal and complete system in $L^{2}((-1,1),d\mu_{\alpha,\beta})$, and it satisfies that

[TABLE]

Along this paper we will work with the operator

[TABLE]

where $I$ denotes the identity operator, instead of $J^{(\alpha,\beta)}$ since the translated operator $-\mathcal{J}^{(\alpha,\beta)}$ is non-negative. In fact, the spectrum of $J^{(\alpha,\beta)}$ is the interval $[-1,1]$, so that the spectrum of $-\mathcal{J}^{(\alpha,\beta)}$ is $[0,2]$.

This paper continues in a natural way the study of harmonic analysis associated with $\mathcal{J}^{(\alpha,\beta)}$ of [1] and [3]. In [1] we carried out an exhaustive analysis of the heat semigroup for $\mathcal{J}^{(\alpha,\beta)}$ and in [3] we investigated the Riesz transform. The main aim of this paper is to study another classical operator in harmonic analysis, the Littlewood-Paley-Stein $g_{k}$-function.

For an appropriate sequence $\{f(n)\}_{n\in\mathbb{N}}$ and $t>0$, the heat semigroup associated with $\mathcal{J}^{(\alpha,\beta)}$ is defined by the identity

[TABLE]

where

[TABLE]

Then, the Littlewood-Paley-Stein $g_{k}^{(\alpha,\beta)}$-functions in this context are given by

[TABLE]

The history of $g$-functions goes back to the seminal paper by J. E. Littlewood and R. E. A. C. Paley [12], published in 1936, where they introduced the $g$-function for the trigonometric Fourier series. The extension to the Fourier transform on $\mathbb{R}^{n}$ was given by E. M. Stein in [17] more than twenty years later. He himself treated the question in a very abstract setting in [18]. In the last few years, there has been a deep research of these operators in different contexts and considering weights. For example, for the Hankel transform they were studied in [6], for Jacobi expansions in [13], for Laguerre expansions in [14], for Hermite expansions in [19], and for Fourier-Bessel expansions in [8].

Our work on discrete harmonic analysis related to Jacobi polynomials pretends to be a generalization of the work in [7] for the discrete Laplacian

[TABLE]

and in [5] in the case of ultraspherical expansions, which corresponds with the case $\alpha=\beta=\lambda-1/2$ of $J^{(\alpha,\beta)}$. In both cases the corresponding $g_{k}$-functions were analysed (in [7] only for $k=1$).

To present our main result we need to introduce some notation. A weight on $\mathbb{N}$ will be a strictly positive sequence $w=\{w(n)\}_{n\geq 0}$. We consider the weighted $\ell^{p}$-spaces

[TABLE]

$1\leq p<\infty$, and we simply write $\ell^{p}(\mathbb{N})$ when $w(n)=1$ for all $n\in\mathbb{N}$.

Furthermore, we say that a weight $w(n)$ belongs to the discrete Muckenhoupt $A_{p}(\mathbb{N})$ class, $1<p<\infty$, provided that

[TABLE]

holds.

The main result of this paper is the following one.

Theorem 1.1**.**

Let $\alpha,\beta\geq-1/2$, $1<p<\infty$, and $w\in A_{p}(\mathbb{N})$. Then,

[TABLE]

where $C_{1}$ and $C_{2}$ are constants independent of $f$.

To prove this theorem we will start by showing that the second inequality in (3) implies the first one. After two appropriate reductions, the former will be deduced from the case $(\alpha,\beta)=(-1/2,-1/2)$ and $k=1$ that we will obtain from discrete Calderón-Zygmund theory.

It is very common to define $g_{k}$-functions in terms of the Poisson semigroup instead of the heat semigroup. In our case the Poisson semigroup can be defined by subordination through the identity

[TABLE]

and then we have the $\mathfrak{g}_{k}^{(\alpha,\beta)}$-function

[TABLE]

The following result will be a consequence of Theorem 1.1.

Corollary 1.2**.**

Let $\alpha,\beta\geq-1/2$, $1<p<\infty$, and $w\in A_{p}(\mathbb{N})$. Then,

[TABLE]

where $C_{1}$ and $C_{2}$ are constants independent of $f$.

We will prove this corollary by controlling the $\mathfrak{g}_{k}^{(\alpha,\beta)}$-function by a finite sum of $g_{k}^{(\alpha,\beta)}$-functions (see Lemma 4.2). This fact will follow from the subordination identity (4).

As an application of Theorem 1.1, we will prove the boundedness of some multipliers of Laplace type for the discrete Fourier-Jacobi series. As it is well known, for each function $f\in L^{2}([-1,1],d\mu_{\alpha,\beta})$ its Fourier-Jacobi coefficients are given by

[TABLE]

and

[TABLE]

where the equality holds in $L^{2}([-1,1],d\mu_{\alpha,\beta})$. Moreover, $\{c_{m}^{(\alpha,\beta)}(f)\}_{m\geq 0}$ is a sequence in $\ell^{2}(\mathbb{N})$. Conversely, for each sequence $f\in\ell^{2}(\mathbb{N})$, the function

[TABLE]

belongs to $L^{2}([-1,1],d\mu_{\alpha,\beta})$ and the Parseval’s identity

[TABLE]

holds. Moreover, $c_{m}^{(\alpha,\beta)}(F_{\alpha,\beta})=f(m)$.

Note that an obvious consequence of (6) is the useful relation

[TABLE]

where $F_{\alpha,\beta}$ is given by (5) and $G_{\alpha,\beta}$ is defined in a similar way.

Given a bounded function $M$ defined on $[0,2]$, the multiplier associated with $M$ is the operator, initially defined on $\ell^{2}(\mathbb{N})$, by the identity

[TABLE]

We say that $T_{M}$ is a Laplace type multiplier when

[TABLE]

with $a$ being a bounded function. From a spectral point of view, $T_{M}=M(\mathcal{J}^{(\alpha,\beta)})$.

The Laplace type multipliers were introduced by Stein in [18, Ch. 2]. There, it is observed that they verify $|x^{k}M^{(k)}(x)|\leq C_{k}$ for $k=0,1,\dots$, and then form a subclass of Marcinkiewicz multipliers. For the operators $T_{M}$ we have the following result.

Theorem 1.3**.**

Let $\alpha,\beta\geq-1/2$, $1<p<\infty$, and $w\in A_{p}(\mathbb{N})$. Then,

[TABLE]

where $C$ is a constant independent of $f$.

From the identity,

[TABLE]

we deduce the following corollary.

Corollary 1.4**.**

Let $\alpha,\beta\geq-1/2$, $1<p<\infty$, and $w\in A_{p}(\mathbb{N})$. Then,

[TABLE]

where $C$ is a constant independent of $f$.

The rest of the paper is organized as follows. Section 2 contains the proof of Theorem 1.1 which relies on a transplantation theorem and the Calderón-Zygmund theory. The proofs of two propositions that are necessary to apply the Calderón-Zygmund theory are provided in Section 3. Section 4 and Section 5 contain the proofs of Corollary 1.2 and Theorem 1.3, respectively.

2. Proof of Theorem 1.1

We consider the Banach space $\mathbb{B}_{k}=L^{2}\left((0,\infty),t^{2k-1}\,dt\right)$, with $k\geq 1$, and the operator

[TABLE]

with

[TABLE]

Then, it is clear that

[TABLE]

A first tool to prove Theorem 1.1 is the following result about the $\ell^{2}$-boundedness of the $g_{k}^{(\alpha,\beta)}$-functions.

Lemma 2.1**.**

Let $\alpha,\beta\geq-1/2$ and $k\geq 1$. Then,

[TABLE]

Proof.

For a sequence $f\in\ell^{2}(\mathbb{N})$, it is satisfied that

[TABLE]

Then, by using (6), we have

[TABLE]

and the proof is completed. ∎

Now, let us see that

[TABLE]

implies the reverse inequality

[TABLE]

Polarising the identity (8), we have

[TABLE]

and, obviously,

[TABLE]

Taking $h(n)=w^{1/p}(n)f_{1}(n)$, we have

[TABLE]

where $w^{\prime}=w^{-1/(p-1)}$ and $p^{\prime}$ is the conjugate exponent of $p$; i.e., $1/p+1/p^{\prime}=1$. Note that $w\in A_{p}(\mathbb{N})$ implies $w^{\prime}\in A_{p^{\prime}}(\mathbb{N})$ and, by (9),

[TABLE]

So, we obtain that

[TABLE]

and taking the supremum over all $f_{1}\in\ell^{p^{\prime}}(\mathbb{N})$ such that $\|f_{1}\|_{\ell^{p^{\prime}}(\mathbb{N})}\leq 1$, we conclude the inequality (10).

In this way, we have reduced the proof of Theorem 1.1 to prove (9). Now, we proceed with two new reductions. First, we are going to use a proper transplantation operator to deduce (9) from the case $(\alpha,\beta)=(-1/2,-1/2)$ for $k\geq 1$. Finally, we will see how to obtain (9) for $g_{k}^{(-1/2,-1/2)}$ with $k>1$ from the case $k=1$. These reductions in the proof are inspired by the work in [9].

For $f\in\ell^{2}(\mathbb{N})$ we define the transplantation operator

[TABLE]

where

[TABLE]

This operator was analysed in [2], where an extension of a classical result from R. Askey [4] was given. In fact, it was proved that

[TABLE]

with weights $w\in A_{p}(\mathbb{N})$, and the analogous weak inequality from $\ell^{1}(\mathbb{N},w)$ into $\ell^{1,\infty}(\mathbb{N},w)$ for weights in the $A_{1}(\mathbb{N})$ class. By a result due to Krivine (see [11, Theorem 1.f.14]), it is possible to give, in an obvious way, a vector-valued extension of the transplantation operator to the space $\mathbb{B}_{k}$, denoted by $\overline{T}_{\alpha,\beta}^{\gamma,\delta}$, satisfying

[TABLE]

with weights in $A_{p}(\mathbb{N})$.

In this way, we have

[TABLE]

Indeed, we have

[TABLE]

and, by using (7) and the identities

[TABLE]

and

[TABLE]

we deduce that

[TABLE]

Applying a similar argument to the other composition the proof of (11) follows.

Now, let us see that it is enough to analyse the $g_{1}^{(-1/2,-1/2)}$-function. In fact, using induction we can deduce the boundedness of the $g_{k}^{(-1/2,-1/2)}$-functions for $k>1$. Let us suppose that the operator $G_{t,k}^{(-1/2,-1/2)}$ is bounded from $\ell^{p}(\mathbb{N},w)$ into $\ell^{p}_{\mathbb{B}_{k}}(\mathbb{N},w)$. Taking $k=1$ and applying again Krivine’s theorem, we deduce that the operator $\overline{G}^{(-1/2,-1/2)}_{t,1}:\ell^{p}_{\mathbb{B}_{k}}(\mathbb{N},w)\longrightarrow\ell^{p}_{\mathbb{B}_{k}\times\mathbb{B}_{1}}(\mathbb{N},w)$, given by

[TABLE]

is bounded. Moreover, $\overline{G}_{t,1}^{(-1/2,-1/2)}\circ G_{s,k}^{(-1/2,-1/2)}$ is a bounded operator from $\ell^{p}(\mathbb{N},w)$ into $\ell^{p}_{\mathbb{B}_{k}\times\mathbb{B}_{1}}(\mathbb{N},w)$. Now, using the identity

[TABLE]

we have

[TABLE]

Finally, to complete the proof of Theorem 1.1 we have to prove (9) for $(\alpha,\beta)=(-1/2,-1/2)$ and $k=1$. This fact will be a consequence of the following propositions.

Proposition 2.2**.**

Let $n,m\in\mathbb{N}$ with $n\not=m$. Then,

[TABLE]

Moreover

[TABLE]

Proposition 2.3**.**

Let $n,m\in\mathbb{N}$ with $n\not=m$. Then,

[TABLE]

and

[TABLE]

The proof of these propositions is the most delicate part of the proof of Theorem 1.1, so it is postponed to the next section.

Now, using the decomposition

[TABLE]

we can apply (12) of Proposition 2.2, Proposition 2.3, and Lemma 2.1 to deduce from the Calderón-Zygmund theory the inequality

[TABLE]

and (13) to obtain that

[TABLE]

finishing the proof of Theorem 1.1.

3. Proof of Proposition 2.2 and Proposition 2.3

Denoting by $T_{n}$ the Chebyshev polynomials, we have

[TABLE]

for $n\not=0$ and where $x=\cos\theta$, and $p_{0}^{(-1/2,-1/2)}(x)=\sqrt{\frac{1}{\pi}}T_{0}(x)=\sqrt{\frac{1}{\pi}}$. Then, the identity (see [16, p. 456])

[TABLE]

where $I_{m}$ denotes the Bessel function of imaginary argument of order $m$, implies

[TABLE]

[TABLE]

To simplify notation, we set $K_{t}(n)=e^{-t}I_{n}(t)$.

We note that the proofs of Propositions 2.2 and 2.3 are similar to the one given in [7, Proposition 4] but we have included them for a self-contained exposition of the paper and to fix some details.

Proof of Proposition 2.2.

The identity [15, eq. (10.29.1)]

[TABLE]

yields

[TABLE]

and

[TABLE]

The next identity is known as Schläfli’s integral representation of Poisson type for modified Bessel functions (see [10, eq. (5.10.22)]):

[TABLE]

Integrating by parts once and twice in (18), we have, respectively, the identities

[TABLE]

and

[TABLE]

Then from (16), using (18), (19), and (20) with $\nu=n-1$, $\nu=n$, and $\nu=n+1$, respectively, we deduce that for $n\geq 1$

[TABLE]

where

[TABLE]

and

[TABLE]

Now, for $n\geq 2$,

[TABLE]

where in the last step, we have applied the change of variables $s=2u-1$ and $r=2v-1$, and

[TABLE]

In a similar way and again for $n\geq 2$, we obtain that

[TABLE]

and

[TABLE]

Hence,

[TABLE]

Now, we prove that

[TABLE]

By Minkowski’s integral inequality, it is clear that

[TABLE]

Similarly, we obtain that $\left\|\frac{\partial K_{t}(1)}{\partial t}\right\|_{\mathbb{B}_{1}}\leq C$ and the proof of (22) is finished.

Finally, using (14), (15), (21), (22), and the identity

[TABLE]

we conclude the proof of the proposition. ∎

Proof of the Proposition 2.3.

By using (14), (15), and (23), the proof will follow from the estimate

[TABLE]

Using (16), we have

[TABLE]

Integrating by parts three times in (18) gives

[TABLE]

Then, using (18), (19), (20), and (25) with $\nu=n-1$, $\nu=n$, $\nu=n+1$, and $\nu=n+2$, respectively, (3) becomes

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

To estimate these inequalities we proceed as in the previous proposition. In fact, for $n\geq 4$,

[TABLE]

and

[TABLE]

[TABLE]

and

[TABLE]

and finally,

[TABLE]

and

[TABLE]

We deduce (24) from the previous estimates for $n\geq 4$. The remainder cases can be proved as (22) in the previous proposition and then,

[TABLE]

4. Proof of Corollary 1.2

First, it is easy to check that

[TABLE]

with

[TABLE]

Then, we have the following result for the $\mathfrak{g}_{k}^{(\alpha,\beta)}$-function which is the analogue of Lemma 2.1.

Lemma 4.1**.**

Let $\alpha,\beta\geq-1/2$ and $k\geq 1$. Then

[TABLE]

This lemma can be proved following step by step the proof of Lemma 2.1, so we omit the details. Now, using polarization, we deduce the identity

[TABLE]

From this fact, we obtain the inequality

[TABLE]

from the direct inequality

[TABLE]

as we did in the proof of Theorem 1.1. Finally, inequality (27) is an immediate consequence of the following lemma.

Lemma 4.2**.**

Let $\alpha,\beta>-1$, then

[TABLE]

where $A_{j}$ are some constants and $[\cdot]$ denotes the floor function.

Proof.

First, we observe that

[TABLE]

for some constants $B_{j}$. Then, from (4), we have

[TABLE]

and, by Minkowsk’s integral inequality,

[TABLE]

where

[TABLE]

Now, by using an appropriate change of variables, we have

[TABLE]

and the result follows. ∎

5. Proof of Theorem 1.3

We need only prove that

[TABLE]

since by Theorem 1.1 we get that

[TABLE]

Moreover, it is enough to prove (28) for sequences in $c_{00}$, the space of sequences having a finite number of non-null terms. First, we have

[TABLE]

which is an elementary consequence of the relation

[TABLE]

Then, applying the semigroup property of $W_{t}^{(\alpha,\beta)}$ we obtain

[TABLE]

and hence,

[TABLE]

In this way,

[TABLE]

Finally,

[TABLE]

and the proof of (28) is completed.