A note on the off-diagonal Muckenhoupt-Wheeden conjecture
We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calderón-Zygmund operators. Namely, given $1 < p < q < \infty$ and a pair of weights $(u; v)$, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: $$ M:L^p(v) \to L^q(u) \quad \mbox{and} \qua...
| Autores: | , , |
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| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2016 |
| País: | España |
| Recursos: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/300 |
| Acesso em linha: | http://hdl.handle.net/20.500.11824/300 |
| Access Level: | acceso embargado |
| Palavra-chave: | Haar shift operators Calderón-Zygmund operators two-weight inequalities testing conditions |
| Resumo: | We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calderón-Zygmund operators. Namely, given $1 < p < q < \infty$ and a pair of weights $(u; v)$, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: $$ M:L^p(v) \to L^q(u) \quad \mbox{and} \quad M: L^{q'} (u^{1-q'}) \to (v^{1-p'} ); $$ then any Calderón-Zygmund operator $T$ and its associated truncated maximal operator $T_{*}$ are bounded from $M:L^p(v)$ to $L^q(u)$. Additionally, assuming only the second estimate for $M$ then $T$ and $T_{*}$ map continuously $M:L^p(v)$ to $L^{q,\infty}(u)$ We also consider the case of generalized Haar shift operators and show that their off-diagonal two weight estimates are governed by the corresponding estimates for the dyadic Hardy-Littlewood maximal function. |
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