Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Boundedness of differential transforms for one-sided fractional Poisson-type operator sequence (1907.07422v2)

Published 17 Jul 2019 in math.CA

Abstract: In this paper, we analyze the convergence speed of a series related with $\mathcal{P}\tau\alpha f$ by discussing the behavior of the family of operators \begin{equation*} T_N\alpha f(t) = \sum{j=N_1}{N_2} v_j(\mathcal{P}{a{j+1}}\alpha f(t)-\mathcal{P}{a_j}\alpha f(t)),\quad ~N=(N_1,N_2)\in \mathbb{Z}2\quad \hbox{with} \quad N_1<N_2, \end{equation*} where ${v_j}{j\in \mathbb Z}$ is a bounded number sequence, and ${a_j}{j\in \mathbb{Z}}$ is a $\rho$-lacunary sequence of positive numbers, that is, $1<\rho \leq a{j+1}/a_j, \text{for all}\ j\in \mathbb{Z}.$ We shall show the boundedness of the maximal operator \begin{equation*}T*f(t)=\sup_N |T_N\alpha f(t)|, \quad t\in\mathbb{R}, \end{equation*} in the one-sided weighted Lebesgue spaces $Lp(\mathbb{R},\omega)(\omega \in A_p-$), $1< p < \infty$. As a consequence we infer the existence of the limit, in norm and almost everywhere, of the family $T_N\alpha f$ for functions in $Lp(\mathbb{R},\omega)$. Results for $L1(\mathbb{R},\omega)(\omega \in A_1-)$, $L\infty(\mathbb{R})$ and $BMO(\mathbb{R})$ are also obtained. It is also shown that the local size of $T*f$, for functions $f$ having local support, is the same with the order of a singular integral. Moreover, if ${v_j}_{j\in \mathbb Z}\in \ellp(\mathbb Z)$, we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.

Summary

We haven't generated a summary for this paper yet.