Weighted norm inequalities for convolution-type operators and one-sided maximal function

Victor Jacob Olesen, Purdue University

Abstract

The classical approach to the study of convergence of approximate identity operators has strong connections to the weighted norm inequalities satisfied by the Hardy-Littlewood maximal function. In particular, if $\phi:\IR\sp{n}\to\IR\sb+,\ \Vert\phi\Vert\sb1=1$ and $\phi\sb{\varepsilon}(x)=\varepsilon\sp{-n}\phi (\varepsilon\sp{-1}x),$ then $\phi\sb{\varepsilon}\* f\to f$ in $L\sp{p},\ 1\le p\le\infty$. Further, if the associated maximal operator$$T\sp*f(x)=\sup\limits\sb{\varepsilon>0} \vert\phi\sb\varepsilon\* f(x)\vert$$is dominated by the the Hardy-Littlewood maximal function, then $\phi\sb{\varepsilon}\* f(x)\to f(x)$ for a.e. x. In Chapter 1, we study convergence questions of the more general convolution-type operators:$$T\sb{\delta}f(x)=\int\sb{\IR\sp{n}}\phi\sb{\delta}(x,t)f(t)d\nu (t).$$Here, $\nu$ is a measure on $\IR\sp{n}$ and $\{\phi\sb{\delta}:\IR\sp{n}\times\IR\sp{n}\to\IR\sb{+}\},\ \delta\in\Gamma$ is an arbitrary collection of measurable functions. We give "weight" conditions on the measures $\mu,\ \nu$ and specify the sequences $\{\delta\sb{ix}\}\subset\Gamma$ to obtain pointwise and norm-convergence of the sequence$$\{ T\sb{\delta\sb{ix}}f(x)\}\sb{i\ge 1}.$$ In Chapter 2, we extend some results of Carleson, Gatto, Gutierrez, Jones, Neugebauer, Rubio de Francia, Xing-Min, and Young to $A\sbsp{p}{+}$ weights. First, given a pair of weights (u, v) on $\IR,$ we ask under what conditions is it possible to find a function w with $w\in A\sbsp{p}{+}$ and $c\sb1u\le w\le c\sb2v$. Next, the question of factorization of a pair $(u,\ v)\in A\sbsp{p}{+}$ is studied and we extend the single weight factorization result of Martin-Reyes, Salvador, and De La Torre. Finally, given a weight w, it is natural to ask under what conditions is it possible to find a weight v so that $v\in A\sbsp{p}{+}$ and either $w(x)\le cv(x)$ or $v(x)\le cw(x)$.

Degree

Ph.D.

Advisors

Neugebauer, Purdue University.

Subject Area

Mathematics

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