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Weighted estimates for tangential boundary behaviour
V. G. Krotov, L. V. Smovzh Belarusian State University, Faculty of Mathematics and Mechanics
Abstract:
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.
Keywords:
space of homogeneous type, tangential boundary behaviour, weighted inequalities.
Received: 30.01.2006
Citation:
V. G. Krotov, L. V. Smovzh, “Weighted estimates for tangential boundary behaviour”, Sb. Math., 197:2 (2006), 193–211
Linking options:
https://www.mathnet.ru/eng/sm1511https://doi.org/10.1070/SM2006v197n02ABEH003753 https://www.mathnet.ru/eng/sm/v197/i2/p57
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Abstract page: | 505 | Russian version PDF: | 195 | English version PDF: | 6 | References: | 57 | First page: | 2 |
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