Boundedness of differential transforms for heat semigroups generated by fractional Laplacian

TL;DR

This paper investigates the boundedness of differential transforms associated with heat semigroups generated by fractional Laplacians, establishing their behavior in $L^p$ and BMO spaces, and analyzing maximal operators and local size properties.

Contribution

It introduces new boundedness results for differential transforms of fractional heat semigroups and analyzes their maximal operators and local size characteristics.

Findings

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

Boundedness of the maximal operator $T^*$.

Local size of maximal differential transforms matches singular integral order.

Abstract

In this paper we analyze the convergence of the following type of series \begin{equation*} T_N f(x)=\sum_{j=N_1}^{N_2} v_j\Big(e^{-a_{j+1}(-\Delta)^\alpha} f(x)-e^{-a_{j}(-\Delta)^\alpha} f(x)\Big),\quad x\in \mathbb R^n, \end{equation*} where $\{e^{-t(-\Delta)^\alpha} \}_{t>0}$ is the heat semigroup of the fractional Laplacian $(-\Delta)^\alpha,$ $N=(N_1, N_2)\in \mathbb Z^2$ with $N_1<N_2,$ $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^p(\mathbb{R}^n)$ and in $BMO(\mathbb{R}^n)$, of the operators $T_N$ and its maximal operator $\displaystyle T^*f(x)= \sup_N |T_N f(x)|.$ 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.