Grigor Karagulyan | Searcharxiv

arXiv2004

An Estimate of the Maximal Operators Associated with Generalized Lacunary Sets

Let $Ω$ be any set of directions (unit vectors) on the plane. In this paper we study maximal operator of the one dimensional maximal function computed in the directions of $Ω$ We are interested in extensions of lacunary sets of directions, to collections we call $N$--lacunary, for integers $N$. We proceed by induction. Say that $Ω$ is 1--lacunary iff $Ω$ is an ordinary lacunary set of vectors. Every $N+1$--lacunary set can be obtained from some $N$--lacunary $Ω_N$ adding some points to $Ω_N$. Between each two neighbor points $a,b\inΩ_N$ we can add a 1--lacunary sequence (finite or infinite). We show that for all $N$ lacunary sets $Ω$, $$ \|M_Ωf(x)\|_2\lesssim{}N \|f\|_2. $$ Observe that every set $Ω$ of $N$ points is $(C\log N)$--lacunary. We then obtain a Theorem of N. Katz \cite{Katz2}. Both the current inequality, and Katz' result are consequence of a general result of Alfonseca, Soria, and Vargas \cites{ASV2}. We offer the current proof as a succinct, self--contained approach to this inequality.

math.CA↗

arXiv2004

Rubio de Francia Littlewood Paley Inequalities and Directional Maximal Functions

In $R^d$, define a maximal function in the directions $v\in \directions\subset\{x \mid \abs x=1\}$ by $$ M^\directions f(x)=\sup_{v\in\directions} \sup_{\zve} \int_{-\ze}^\ze \abs{f(x-vy)} dy. $$ For a function $f$ on $\ZR^d$, let $S_\zw f$ denote the Fourier restriction of $f$ to a region $\zw$. We are especially interested taking \zw to be a sector of $R^d$ with base points at the origin. A sector is a product of the interval $(0,\infty)$ with respect to a choice of (non orthogonal) basis. What is most important is that the basis is a subset of $\directions$. Consider a collection $\zW$ of pairwise disjoint sectors $\zw$ as above. Assume that $M^\directions $ maps $L^p$ into $L^p$, for some $1<p<\zI $. Then we have the following Littlewood--Paley inequality $$ \NORm \Bigl[\sum_{\zw\in\zW}\abs{S_\zw f}^2\Bigr]^{1/2}.q.\lesssim{}\norm f.q., \qquad 2\le q<2 \frac p{p-1}. $$ The one dimensional analogue of this inequality is due to Rubio de Francia. The conclusion when the set of vectors is a fixed basis is known, is due to Journé. Our method of proof relies on a phase plane analysis. We introduce a notion of Carleson measures adapted to $\directions$, and demonstrate a John Nirenberg inequality for these measures. The John Nirenberg inequality, and an obvious $L^2$ estimate will prove the Theorem.

math.CA↗