Boundedness of differential transforms for heat semigroups generated by fractional Laplacian

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}(-Δ)^α} f(x)-e^{-a_{j}(-Δ)^α} 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 $T^{\ast }f(x)=\underset{N}{\sup }|T_{N}f(x)|.$ 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.

18 pages

References in corpus (2)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations
• Boundedness of differential transforms for one-sided fractional Poisson-type operator sequence