The maximal number of limit cycles in a family of polynomial systems

关于一类强分数极大算子的家族

On a family of strong fractional maximal operators

本文研究了一类参数化的强分数极大算子，证明了其从$ L^p $到$ L^q $的有界性，其中$ 1<p\le q<\infty $。

On Families between the Hardy-Littlewood and Spherical maximal functions

We study a family of maximal operators that provides a continuous link connecting the Hardy-Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents.

Kakeya-type sets for Geometric Maximal Operators

Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $λ$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($λ$ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < $\infty$. Precisely, if $Ω$ is an arbitrary set of angles in [0, $π$ 4), we prove that any rarefied basis B of the directional basis R $Ω$ yields an operator MB that has the same L p-behavior th…

The discrete spherical averages over a family of sparse sequences

We initiate the study of the $\ell^p(\mathbb{Z}^d)$-boundedness of the arithmetic spherical maximal function over sparse sequences. We state a folklore conjecture for lacunary sequences, a key example of Zienkiewicz and prove new bounds for a family of sparse sequences that achieves the endpoint of the Magyar--Stein--Wainger theorem for the full discrete spherical maximal function. Perhaps our most interesting result is the boundedness of a discrete spherical maximal function in $\mathbb{Z}^4$ over an infinite, albeit sparse, set of radii. Our methods include the Kloosterman refinement for the…

A note on the maximal operator on weighted Morrey spaces

In this paper we consider weighted Morrey spaces ${\mathcal M}_{λ, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{λ, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal F}}\left(\frac{1}{|Q|^λ}\int_Q|f|^pw\right)^{1/p},$$ and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy--Littlewood maximal operator on ${\mathcal M}_{λ, {\mathcal F}}^p(w)$. In the case of the global Morrey spaces (when ${\mathcal F}$ is the family of all cubes in ${\mathbb R}^n$) this question is still open. In the case of t…