Mathematics – Classical Analysis and ODEs
Scientific paper
2004-04-02
Mathematics
Classical Analysis and ODEs
12 pages
Scientific paper
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
Karagulyan Grigor
Lacey Michael T.
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