Weighted estimates for tangential boundary behaviour

V. G. Krotov ; L. V. Smovzh

Sbornik. Mathematics, Tome 197 (2006) no. 2, pp. 193-211

Résumé

Let $(X,\mu,d)$ be a space of homogeneous type (here $d$ is a quasimetric and $\mu$ a measure). A function $\varepsilon$ of modulus of continuity kind gives rise to approach regions $\Gamma_{\varepsilon}(x)$ at the boundary of $\mathbf{X}$, $\mathbf{X}=X\times[0,1)$, where for a point $x\in X$, $$ \Gamma_{\varepsilon}(x)=\{(y,t)\in\mathbf{X}:d(x,y)\varepsilon(1-t)\}. $$ These are ‘tangential’ regions if $\lim_{t\to+0}\varepsilon(t)/t=\infty$. Weighted $L^p$-estimates are proved for the corresponding maximal functions of integral operators. Applications of these estimates to potentials in $\mathbb{R}^n$ and to multipliers of homogeneous expansions of holomorphic functions in the Hardy classes in the unit ball of $\mathbb{C}^n$ are presented. Bibliography: 20 titles.

Détail

Keywords: space of homogeneous type, tangential boundary behaviour, weighted inequalities.