Bellman Functions and Dimension Free $L^p$estimates for the Riesz Transforms in Bessel settings

TL;DR

This paper proves that Riesz transforms related to Bessel operators are bounded on L^p spaces independently of dimension, providing explicit p-dependent estimates using Bellman functions.

Contribution

It introduces a Bellman function approach to establish dimension-free L^p bounds for Bessel-related Riesz transforms with explicit p-dependence.

Findings

Dimension-free L^p bounds for Bessel-Riesz transforms.

Linear growth of L^p norms with respect to p.

Explicit estimates of the L^p norms in terms of p.

Abstract

In this article we prove dimension free $L^p$-boundedness of Riesz transforms associated with a Bessel diferential operator. We obtain explicit estimates of the $L^p$-norms for the Bessel-Riesz transforms in terms of p, establishing a linear behaviour with respect to p. We use the Bellman function technique to prove a bilinear dimension free inequality involving Poisson semigroups defined through this Bessel operator.

Equations

$$R_{j}f(x)=\frac{\Gamma (\frac{d+1}{2})}{{\pi }^{\frac{d+1}{2}}}\underset{ϵ\to 0^{+}}{\lim }{\int }_{|x-y|>ϵ}\frac{x_j-y_j}{|x-y{|}^{d+1}}f(y)dy,a.e.x=(x_1,\dots ,x_d)\in {\mathbb{R}}^d.$$

$$\|\mathcal{H}\|_{L^{p}(\mathbb R^{d})\rightarrow L^{p}(\mathbb R^{d})}=\left\{\begin{array}[]{lc}\tan(\frac{\pi}{2p}),&1<p\leq 2,\\ \cot(\frac{\pi}{2p}),&p\geq 2.\end{array}\right.$$

$$B_{\alpha }=-x^{2\alpha }\frac{d}{dx}{x}^{-2\alpha }\frac{d}{dx}=-\frac{d^2}{d{x}^2}+\frac{2\alpha }{x}\frac{d}{dx},$$

$$h_{\alpha }f(x)={\int }_0^{\infty }{\varphi }_y^{\alpha }(x)f(y)x^{2\alpha }dx,x\in (0,\infty ),$$

$$h_{\alpha }(B_{\alpha }f)(x)=x^2{h}_{\alpha }(f)(x),x\in (0,\infty )$$

$$R_{\alpha }f(x)=-x{h}_{\alpha +1}\left(\frac{1}{y}{h}_{\alpha }f(y)\right)(x).$$

$$B_{\alpha }^{-1/2}f=h_{\alpha }\left(\frac{1}{y}{h}_{\alpha }f(y)\right).$$

$$R_{\alpha }f(x)=\frac{d}{dx}{B}_{\alpha }^{-1/2}f(x),x\in (0,\infty ).$$

$$R_{\alpha }f(x)=\underset{ϵ\to 0^{+}}{\lim }{\int }_{|x-y|>ϵ}{R}_{\alpha }(x,y)f(y)y^{2\alpha }dy,a.e.x\in (0,\infty ),$$

$$R_{\alpha }(x,y)={\int }_0^{\infty }{\partial }_x{\mathcal{P}}_t^{\alpha }(x,y)dt,x,y\in (0,\infty ),x\ne y,$$

$$B_{\alpha }=\sum _{i=1}^d{B}_{{\alpha }_i,x_i}=\sum _{i=1}^d\left(-\frac{{\partial }^2}{\partial x_i^2}+\frac{2{\alpha }_i}{x_i}\frac{\partial }{\partial x_i}\right).$$

$$h_{\alpha }f(x)={\int }_{{\mathbb{R}}_{+}^d}{\varphi }_y^{\alpha }(x)f(y)y^{2\alpha }dy,$$

$${\varphi }_y^{\alpha }(x)=\prod _{j=1}^d(x_j{y}_j{)}^{-{\alpha }_j+1/2}{J}_{{\alpha }_j-1/2}(x_j{y}_j),x=(x_1,\dots ,x_d),y=(y_1,\dots ,y_d)\in {\mathbb{R}}_{+}^d,$$

$$R_{\alpha ,i}f(x)=-x_i{h}_{\alpha +e_i}\left(\frac{1}{|y|}{h}_{\alpha }f(y)\right)(x),x=(x_1,\dots ,x_d)\in {\mathbb{R}}_{+}^d,$$

$${\gamma }_{m,k}(\varphi )=:\underset{x\in {\mathbb{R}}_{+}^d}{\sup }|x{|}^m\left|\frac{{\partial }^{k_1+\dots +k_d}}{\partial x_1\dots \partial x_d}\varphi (x)\right|<\infty .$$

$$h_{\alpha }(B_{\alpha }\varphi )(x)=|x{|}^2{h}_{\alpha }(\varphi )(x),x\in {\mathbb{R}}_{+}^d.$$

$${\mathcal{A}}_{\alpha ,c}({\mathbb{R}}_{+}^d)={\mathcal{A}}_{{\alpha }_1,c}({\mathbb{R}}_{+})\otimes \dots \otimes {\mathcal{A}}_{{\alpha }_d,c}({\mathbb{R}}_{+}).$$

$$B_{\alpha }^{-1/2}(\varphi )=h_{\alpha }\left(\frac{1}{|y|}{h}_{\alpha }(\varphi )\right)$$

$$R_{\alpha ,i}\varphi ={\partial }_{x_i}{B}_{\alpha }^{-1/2}(\varphi ),\varphi \in {\mathcal{A}}_{\alpha ,c}({\mathbb{R}}_{+}^d).$$

$$R_{\alpha }={\left(\sum _{i=1}^d|R_{\alpha ,i}{|}^2\right)}^{1/2}.$$

$$\Vert R_{\alpha }f{\Vert }_{L^p({\mathbb{R}}_{+}^d,x^{2\alpha }dx)}\le 48(p^{\ast }-1)\Vert f{\Vert }_{L^p({\mathbb{R}}_{+}^d,x^{2\alpha }dx)},$$

$$C_{\alpha ,k}=\{\text{compositions of }k\text{ operators among }{R}_{\alpha ,1},\dots ,R_{\alpha ,d}\}.$$

$${\Vert {\left(\sum _{R\in C_{\alpha ,k}}|Rf{|}^2\right)}^{1/2}\Vert }_{L^p({\mathbb{R}}_{+}^d,x^{2\alpha }dx)}\le 48^k(p^{\ast }-1{)}^k\Vert f{\Vert }_{L^p({\mathbb{R}}_{+}^d,x^{2\alpha }dx)}.$$

$$\Vert R_{\alpha ,i}f{\Vert }_{L^p({\mathbb{R}}_{+}^d,x^{2\alpha }dx)}\le 48(p^{\ast }-1)\Vert f{\Vert }_{L^p({\mathbb{R}}_{+}^d,x^{2\alpha }dx)},$$

$$xy$$

$$\alpha -\text{amsbb}1/2$$

$$x^{\alpha }$$

$$J_{\nu }(x)=\prod _{j=1}^d{J}_{{\nu }_j}(x_j),I_{\nu }(x)=\prod _{j=1}^d{I}_{{\nu }_j}(x_j),$$

$${\Delta }_{\alpha ,x}=-\sum _{j=1}^d{\delta }_j^{\ast }{\delta }_j=\sum _{j=1}^d{x}_j^{-2{\alpha }_j}\frac{\partial }{\partial x_j}\left(x_j^{2{\alpha }_j}\frac{\partial }{\partial x_j}\right)=\sum _{j=1}^d\left({\partial }_{x_j}^2-\frac{2{\alpha }_j}{x_j}{\partial }_{x_j}\right),$$

$${\delta }_j^{\ast }=-x_j^{-2{\alpha }_j}\frac{\partial }{\partial x_j}(x_j^{2{\alpha }_j}\cdot ),{\delta }_j=\frac{\partial }{\partial x_j}=:{\partial }_{x_j},$$

$$[{\delta }_j,{\delta }_j^{\ast }]={\delta }_j{\delta }_j^{\ast }-{\delta }_j^{\ast }{\delta }_j=\frac{2{\alpha }_j}{x_j^2}\ge 0.$$

Full text

The first and the third authors are partially supported by MTM2016-79436-P. The second author is partially supported by PIP 2015-900 (CONICET). The fourth author is supported by CAI+D (UNL)

1. Introduction and main results

For every $j=1,\dots,d$, the classical Riesz transform $R_{j}f$ of $f\in L^{p}(\mathbb R^{d})$, $1\leq p<\infty$, is defined by

[TABLE]

As it is well-known, $R_{j}$ is bounded on $L^{p}(\mathbb R^{d})$ for every $1<p<\infty$ and $j=1,\dots,d$ with constant independent of dimension. Stein [39, Theorem, p. 71] proved something stronger than that: for every $1<p<\infty$, there exists $A_{p}>0$ such that $\|\mathbf{R}f\|_{L^{p}(\mathbb R^{d})}\leq A_{p}\|f\|_{L^{p}(\mathbb R^{d})}$ for $f\in L^{p}(\mathbb R^{d})$, where $\mathbf{R}f=\left(\sum_{j=1}^{d}|R_{j}f|^{2}\right)^{1/2}$. By using transference methods, Duoandikoetxea and Rubio de Francia [15] gave another proof of Stein’s result.

Let $\mathcal{H}$ denote the Hilbert transform on $\mathbb R$. Gokhberg and Krupnik [17] and Pichorides [35] proved that

[TABLE]

Iwaniev and Martin [22] extended this result to higher dimensions proving that, for every $j=1,\dots,d$, $\|R_{j}\|_{L^{p}(\mathbb R^{d})\rightarrow L^{p}(\mathbb R^{d})}=\|\mathcal{H}\|_{L^{p}(\mathbb R^{d})\rightarrow L^{p}(\mathbb R^{d})}$, $1<p<\infty$. This result was also showed by Bañuelos and Wang [3] using probabilistic methods.

This phenomenon of dimension free estimates has been observed for Riesz transforms appearing in other settings. For instance, Coulhon, Muller and Zienkiewicz [8] worked with Riesz transforms on the Heisenberg group; Lust-Piquard [24] considered discrete Riesz transforms on a product of Abelian groups and Urban and Zienkiewicz [41] proved these estimates for Riesz transforms associated with Schrö- dinger operators.

In recent years, finding dimension free $L^{p}$–estimates for Riesz transforms associated with orthogonal systems has been on the rise considerably. In the case of Hermite polynomials which are associated with the Ornstein–Uhlenbeck differential operator and the Gaussian measure, this kind of results were obtained by Pisier [36] and Gutiérrez [19] (see also [18] and [28] for proofs using probabilistic methods). Harboure, de Rosa, Segovia and Torrea [20] proved $L^{p}$-dimension free boundedness for Riesz transforms related to the harmonic oscillator and the Hermite functions (see [25] for another proof). In different settings associated with Laguerre operators, the results have been obtained by Nowak [31] and Stempak and Wróbel [40]. A general result that applies to different orthogonal systems was proved by Forzani, Sasso and Scotto [16]. In [16], [19], [20], [25], [31], and [40] a procedure based upon the boundedness of Littlewood-Paley functions is used and it is inspired by the Stein’s method of proof developed in [39].

Recently, Dragičevic and Volberg in a series of papers ([11], [12], [13], and [14]) have developed another approach, different from any previous ones, where they use a technique involving Bellman functions. These Bellman functions allow them to obtain dimension free bilinear Littlewood–Paley estimates. Then, as a consequence, dimension free $L^{p}$-estimates for Riesz transforms are also obtained. Dragičevic and Volberg have studied classical and Gaussian (Hermite polynomial) context in [11]. Riesz transforms associated with the harmonic oscillator and more general Schrödinger operators were analyzed in [12] and [14], respectively. In [13], bilinear Littlewood–Paley estimates related to elliptic operators were established.

By using the Bellman function techniques it is possible to get dimension free $L^{p}$-estimates with better constants than the ones given by other procedures. Furthermore, it is proved that these constants are linear in $\max\{p,p/(p-1)\}$ for $1<p<\infty$.

Bellman functions have also been utilized in the boundedness of Riesz transforms appearing in other settings, such as: Bakry context ([7] and [9]), Hodge–Laguerre operators ([26]), discrete orthogonal systems ([43] and [44]), weighted Riesz transforms ([10]) and Beurling operator ([34]).

In this paper we obtain dimension free $L^{p}$-estimates for Riesz transforms in this Bessel setting by using Bellman functions techniques.

Muckenhoupt and Stein ([29]) started with the study of harmonic analysis associated to Bessel operators. They considered, for $\alpha\geq 0$, the Bessel operator

[TABLE]

and the Hankel transformation $h_{\alpha}$ defined, for every $f\in L^{1}((0,\infty),x^{2\alpha}dx)$, by

[TABLE]

where $\phi_{y}^{\alpha}(x)=(xy)^{-\alpha+1/2}J_{\alpha-1/2}(xy)$ are the eigenfunctions of $B_{\alpha}$, and $J_{\mu}$ represents the Bessel function of first kind and order $\mu$. Notice that $\phi_{y}^{\alpha}(x)=\phi_{x}^{\alpha}(y)$, $x,y\in(0,\infty)$, and $\phi_{y}^{\alpha}\in L^{p}((0,\infty),x^{2\alpha}dx)$ if and only if $\frac{2\alpha+1}{\alpha}<p\leq\infty$, when $\alpha>0$, and $p=\infty$, when $\alpha=0$ (see [23, (5.16.1), p. 134]).

Since the function $z^{-\mu}J_{\mu}(z)$ is bounded on $(0,\infty)$ for $\mu\geq-1/2$ (see [23, (5.16.1), p. 134]), $h_{\alpha}$ defines a bounded operator from $L^{1}((0,\infty),$ $x^{2\alpha}dx)$ to $L^{\infty}((0,\infty),x^{2\alpha}dx)=L^{\infty}((0,\infty),dx)$. Also, $h_{\alpha}$ can be extended from $L^{1}((0,\infty),$ $x^{2\alpha}dx)\cap L^{2}((0,\infty),x^{2\alpha}dx)$ to $L^{2}((0,\infty),x^{2\alpha}dx)$ as an isometry on $L^{2}((0,\infty),x^{2\alpha}dx)$ and $h_{\alpha}^{-1}=h_{\alpha}$ ([21]). Let us note that both the Hankel transformation and the Bessel operator are connected by

[TABLE]

for every $f\in C_{c}^{\infty}(0,\infty)$, the space of smooth functions with compact support in $(0,\infty)$.

Muckenhoupt and Stein ([29]) introduced the Riesz transform $R_{\alpha}$ associated with $B_{\alpha}$ as follows. For every $f\in L^{2}((0,\infty),x^{2\alpha}dx)$,

[TABLE]

Since $h_{\alpha}$ is bounded on $L^{2}((0,\infty),x^{2\alpha}dx)$, it is clear that $R_{\alpha}$ is also a bounded operator on $L^{2}((0,\infty),x^{2\alpha}dx)$. As in the classical case, $R_{\alpha}$ can be extended from $L^{2}((0,\infty),x^{2\alpha}dx)\cap L^{p}((0,\infty),x^{2\alpha}dx)$ to $L^{p}((0,\infty),x^{2\alpha}dx)$ as an $L^{p}$–bounded operator, for every $1<p<\infty$, and from $L^{1}((0,\infty),x^{2\alpha}dx)$ into $L^{1,\infty}((0,\infty),x^{2\alpha}dx)$ ([29]). On the other hand, $L^{p}$-weighted inequalities for $R_{\alpha}$ were established in [2] and, more recently, in [6].

We define $\mathcal{A}_{\alpha,c}(0,\infty)$ as the space consisting of all those functions $\phi\in C^{\infty}(0,\infty)$ $\cap L^{1}((0,\infty),x^{2\alpha}dx)$ such that $h_{\alpha}(\phi)\in C_{c}^{\infty}(0,\infty)$. Since $h_{\alpha}$ is an isometry in $L^{2}((0,\infty),$ $x^{2\alpha}dx)$, the space $\mathcal{A}_{\alpha,c}(0,\infty)$ is a dense subspace of $L^{2}((0,\infty),x^{2\alpha}dx)$.

Motivated by (1.1) we define, for every $f\in\mathcal{A}_{\alpha,c}(0,\infty)$, $B_{\alpha}^{-1/2}f$ by

[TABLE]

Then, we have that for every $f\in\mathcal{A}_{\alpha,c}(0,\infty)$

[TABLE]

Equality (1.2) says that $R_{\alpha}$ is the Riesz transform associated with $B_{\alpha}$ in the sense of Stein [38]. Furthermore, $R_{\alpha}$ can be seen as a principal value integral operator. For every $f\in L^{p}((0,\infty),x^{2\alpha}dx)$, $1\leq p<\infty$, we can write

[TABLE]

where

[TABLE]

and $\mathcal{P}_{t}^{\alpha}$ denotes the Poisson kernel associated to $B_{\alpha}$ (see Section 2 for definitions). The Bessel–Riesz transform $R_{\alpha}$ turns out to be a Calderón–Zygmund operator in the homogeneous type space $((0,\infty),x^{2\alpha}dx,\rho)$ where $\rho$ represents the Euclidean metric on $(0,\infty)$ (see [5]).

Bessel–Riesz transforms in higher dimensions were considered in [4]. Given a multi-index $\alpha=(\alpha_{1},\dots,\alpha_{d})\in{\mathbb R^{d}_{\geq 0}:=[0,\infty)^{d}}$, let us define the $d$-dimensional Bessel differential operator as

[TABLE]

The Hankel transform $h_{\alpha}(f)$ of $f$ is defined, for $x\in\mathbb R^{d}_{+}$, by

[TABLE]

where

[TABLE]

$y^{2\alpha}=\prod_{j=1}^{d}y_{j}^{2\alpha_{j}}$ and $f\in L^{1}(\mathbb R^{d}_{+},x^{2\alpha}dx)$. For every $y\in\mathbb{R}_{+}^{d}$, the function $\phi_{y}^{\alpha}$ is an eigenfunction of $B_{\alpha}$, being $B_{\alpha}\phi_{y}^{\alpha}=|y|^{2}\phi_{y}^{\alpha}$