Two weight function norm inequalities for the Poisson integral
HTML articles powered by AMS MathViewer
- by Benjamin Muckenhoupt PDF
- Trans. Amer. Math. Soc. 210 (1975), 225-231 Request permission
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.References
- Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- Benjamin Muckenhoupt and Richard L. Wheeden, Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform, Studia Math. 55 (1976), no. 3, 279–294. MR 417671, DOI 10.4064/sm-55-3-279-294
- Marvin Rosenblum, Summability of Fourier series in $L^{p}(d\mu )$, Trans. Amer. Math. Soc. 105 (1962), 32–42. MR 160073, DOI 10.1090/S0002-9947-1962-0160073-1
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 210 (1975), 225-231
- MSC: Primary 42A40
- DOI: https://doi.org/10.1090/S0002-9947-1975-0374790-0
- MathSciNet review: 0374790