Borderline Weighted Estimates for Commutators of Singular Integrals
In this paper we establish the following estimate \[ w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > \lambda\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}\Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)M_{L(\log L)^{1+\varepsilon}}w(x)dx \] where $w\geq0, \,...
| Authors: | ; |
|---|---|
| Format: | Article |
| Status: | Accepted version |
| Publication date: | 2016 |
| Country: | España |
| Institution: | Basque Center for Applied Mathematics (BCAM) |
| Repository: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/299 |
| Online Access: | http://hdl.handle.net/20.500.11824/299 |
| Access Level: | Open access |
| Keyword: | Commutators Rubio de Francia Extrapolation $A_p$ weights Hardy-Littlewood maximal function |
| _version_ | 1878738565706285058 |
|---|---|
| author | Pérez, C. |
| author2 | Rivera-Ríos, I.P. |
| author2_role | author |
| author_browse | Pérez, C. Rivera-Ríos, I.P. |
| author_facet | Pérez, C. Rivera-Ríos, I.P. |
| author_role | author |
| collection | BIRD. BCAM's Institutional Repository Data |
| description | In this paper we establish the following estimate \[ w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > \lambda\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}\Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)M_{L(\log L)^{1+\varepsilon}}w(x)dx \] where $w\geq0, \, 0<\varepsilon<1$ and $\Phi(t)=t(1+\log^+(t))$. This inequality relies upon the following sharp $L^p$ estimate \[ \|[b,T]f\|_{L^{p}(w)}\leq c_{T}\left(p'\right)^{2}p^{2}\left(\frac{p-1}{\delta}\right)^{\frac{1}{p'}} \|b\|_{BMO} \, \|f \|_{L^{p}(M_{L(\log L)^{2p-1+\delta}}w)} \]where $1<p<\infty, w\geq0 \text{ and } 0<\delta<1.$ As a consequence we recover the following estimate essentially contained in \cite{MR3008263}: \[w\left(\{x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| >\lambda\}\right)\leq c_T\,[w]_{A_{\infty}}\left(1+\log^{+}[w]_{A_{\infty}}\right)^{2}\int_{\mathbb{R}^{n}} \Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)Mw(x)dx\] We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_b1295fcc31e692bee9266bb55d7a7dde |
| institution | Basque Center for Applied Mathematics (BCAM) |
| instname_str | Basque Center for Applied Mathematics (BCAM) |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:bird.bcamath.org:20.500.11824/299 |
| publishDate | 2016 |
| publishDateSort | 2016 |
| reponame_str | BIRD. BCAM's Institutional Repository Data |
| spelling | Borderline Weighted Estimates for Commutators of Singular Integrals Pérez, C. Rivera-Ríos, I.P. Commutators Rubio de Francia Extrapolation $A_p$ weights Hardy-Littlewood maximal function In this paper we establish the following estimate \[ w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > \lambda\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}\Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)M_{L(\log L)^{1+\varepsilon}}w(x)dx \] where $w\geq0, \, 0<\varepsilon<1$ and $\Phi(t)=t(1+\log^+(t))$. This inequality relies upon the following sharp $L^p$ estimate \[ \|[b,T]f\|_{L^{p}(w)}\leq c_{T}\left(p'\right)^{2}p^{2}\left(\frac{p-1}{\delta}\right)^{\frac{1}{p'}} \|b\|_{BMO} \, \|f \|_{L^{p}(M_{L(\log L)^{2p-1+\delta}}w)} \]where $1<p<\infty, w\geq0 \text{ and } 0<\delta<1.$ As a consequence we recover the following estimate essentially contained in \cite{MR3008263}: \[w\left(\{x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| >\lambda\}\right)\leq c_T\,[w]_{A_{\infty}}\left(1+\log^{+}[w]_{A_{\infty}}\right)^{2}\int_{\mathbb{R}^{n}} \Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)Mw(x)dx\] We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols. http://hdl.handle.net/20.500.11824/299 |
| spellingShingle | Borderline Weighted Estimates for Commutators of Singular Integrals Pérez, C. Commutators Rubio de Francia Extrapolation $A_p$ weights Hardy-Littlewood maximal function |
| status_str | acceptedVersion |
| title | Borderline Weighted Estimates for Commutators of Singular Integrals |
| title_full | Borderline Weighted Estimates for Commutators of Singular Integrals |
| title_fullStr | Borderline Weighted Estimates for Commutators of Singular Integrals |
| title_full_unstemmed | Borderline Weighted Estimates for Commutators of Singular Integrals |
| title_short | Borderline Weighted Estimates for Commutators of Singular Integrals |
| title_sort | Borderline Weighted Estimates for Commutators of Singular Integrals |
| topic | Commutators Rubio de Francia Extrapolation $A_p$ weights Hardy-Littlewood maximal function |
| topic_facet | Commutators Rubio de Francia Extrapolation $A_p$ weights Hardy-Littlewood maximal function |
| url | http://hdl.handle.net/20.500.11824/299 |