Two weight function norm inequalities for the Poisson integral

Abstract:

Let $f(x)$ denote a complex valued function with period $2\pi$, let \[ {P_r}(f,x) = \frac {1}{{2\pi }}\int _{ - \pi }^\pi {\frac {{(1 - {r^2})f(y)dy}}{{1 - 2r\cos (x - y) + {r^2}}}} \] be the Poisson integral of $f(x)$ and let $|I|$ denote the length of an interval $I$. For $1 \leqslant p < \infty$ and nonnegative $U(x)$ and $V(x)$ with period $2\pi$ it is shown that there is a $C$, independent of $f$, such that \[ \sup \limits _{0 \leqslant r < 1} \int _{ - \pi }^\pi {|{P_r}(f,x){|^p}U(x)dx \leqslant C\int _{ - \pi }^\pi {|f(x){|^p}V(x)dx} } \] if and only if there is a $B$ such that for all intervals $I$ \[ \left [ {\frac {1}{{|I|}}\int _I {U(x)dx} } \right ]{\left [ {\frac {1}{{|I|}}\int _I {{{[V(x)]}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1.}} \leqslant B.\] Similar results are obtained for the nonperiodic case and in the case where $U(x)dx$ and $V(x)dx$ are replaced by measures.