Boundedness of differential transforms for fractional Poisson type operators generated by parabolic operators

TL;DR

This paper studies the boundedness and convergence of fractional Poisson-type differential transforms generated by parabolic operators, establishing their behavior in $L^p$ and $BMO$ spaces and analyzing their local size.

Contribution

It introduces new boundedness results for fractional differential transforms associated with parabolic operators and compares their local size to classical singular integrals.

Findings

Boundedness of the operators in $L^p$ and $BMO$ spaces.

Maximal operators are also bounded.

Local size matches that of singular integrals for functions with local support.

Abstract

In this paper we analyze the convergence of the following type of series $$ T_N^\alpha f(x,t)=\sum_{j=N_1}^{N_2} v_j(P_{a_{j+1}}^\alpha f(x,t)-P_{a_j}^\alpha f(x,t)),\quad (x,t)\in \mathbb R^{n+1}, \ N=(N_1, N_2)\in \mathbb Z^2,\ \alpha>0, $$ where $\{P_{\tau}^\alpha \}_{\tau>0}$ is the fractional Poisson-type operators generated by the parabolic operator $L=\partial_t-\Delta$ with $\Delta$ being the classical Laplacian, $\{v_j\}_{j\in \mathbb Z}$ a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ an increasing real sequence. Our analysis will consist {of} the boundedness, in $L^p(\mathbb{R}^n)$ and in $BMO(\mathbb{R}^n)$, of the operators $T^{\alpha}_N$ and its maximal operator $ T^*f(x)= \sup_{N\in \mathbb Z^2} |T^{\alpha}_N f(x)|.$ It is also shown that the local size of the maximal differential transform operators is the same with the order of a singular integral for functions $f$ having local support. Moreover, if $\{v_j\}_{j\in \mathbb Z}\in \ell^p(\mathbb Z)$, we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.