Singular integrals in spaces of functions summable with a monotone weight

E. G. Guseinov

Sbornik. Mathematics, Tome 60 (1988) no. 1, pp. 29-46

Résumé

Inequalities of the form \begin{equation} \int_S|Tu(x)|^p\omega_1(r(x))\,dx\leqslant C\int_S|u(x)|^p\omega(r(x))\,dx \label{1} \end{equation} are studied, where $1<p<\infty$, $\omega$ and $\omega_1$ are positive monotone functions, and $T$ denotes, respectively, a) a multidimensional Calderón–Zygmund singular integral extended over a domain $S$ in $R_m$ ($r(x)$ is the distance from $x\in S$ to the boundary of the domain); and b) the conjugate function ($S=(-\pi,\pi)$, $r(x)=|x|$). In case a) a class of domains is distinguished (domains of type $\alpha$ in $R_m$) which, in particular, contains domains with smooth boundaries; for each domain of type $\alpha$, $0\le\alpha<m$, sufficient conditions are found for the validity of (1), and examples are given which demonstrate their necessity. In case b) we give necessary and sufficient conditions for the validity of (1). For monotone weight functions these results amplify and supplement corresponding investigations by Hunt, Muckenhoupt, and Wheeden (Trans. Amer. Math. Soc. 176 (1973), 227–251) and by Coifman and Fefferman (Studia Math. 51 (1974), 241–250). Bibliography: 32 titles.

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