Hardy–Littlewood–Sobolev-Type Inequality for the Fractional Littlewood–Paley g-Function in Jacobi Analysis

Abstract

We introduce the fractional Littlewood–Paley g-function of order $s, ~~s>0$, noted $g_{s}^{\alpha ,\beta }$, associated with the Jacobi operator $\Delta _{\alpha ,\beta }$ on $(0, \infty$) as the operator

$$\begin{aligned}g_{s}^{\alpha , \beta }(f)(x)=\left( \int _{0}^{\infty }t^{2s+1}|\frac{\partial u_{\alpha , \beta }(f)}{\partial t}(x,t)|^{2}dt\right) ^{\frac{1}{2}}, \end{aligned}$$

where $u_{\alpha , \beta }(f)$ is the Poisson integral defined by $u_{\alpha , \beta }(f)= P_{t}^{\alpha , \beta }*_{\alpha , \beta }f$ ($*_{\alpha , \beta }$ being the convolution in the Jacobi setting). We establish the following Hardy–Littlewood–Sobolev-type inequality: For $0<s<2(\alpha +1),~~1<p<\frac{2(\alpha +1)}{s}$ and $\frac{1}{q}=\frac{1}{p}-\frac{s}{2(\alpha +1)}$, there exists a constant $C_{\alpha , s, p}$ such that for all $f \in L^{p}([0,+\infty [,d\mu _{\alpha ,\beta })$,

$$\begin{aligned} \Vert g_{s}^{\alpha ,\beta }f\Vert _{q,\mu }\le C_{\alpha , s, p} \Vert f\Vert _{p,\mu }. \end{aligned}$$

Next, if $p=1$, $0<s<2(\alpha +1)$ and $q=\frac{\alpha +1}{2(\alpha +1)-s}$, we prove that the operator $g_{s}^{\alpha ,\beta }$ is of weak type (1, q).