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

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$$T\sp*f(x)=\sup\limits\sb{\varepsilon\u3e0} \vert\phi\sb\varepsilon\* f(x)\vert$$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 $v(x)\le cw(x)$