Anisotropic Singular Integrals in Product Spaces
Let \(A_i\) for \(i=1, 2\) be an expansive dilation, respectively, on \({\mathbb R}^n\) and \({\mathbb R}^m\) and \(\vec A\equiv(A_1, A_2)\). Denote by \({\mathcal A}_\infty(\rnm; \vec A)\) the class of Muckenhoupt weights associated with \(\vec A\). The authors introduce a class of anisotropic sing...
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19.07.2010
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| Abstract | Let \(A_i\) for \(i=1, 2\) be an expansive dilation, respectively, on \({\mathbb R}^n\) and \({\mathbb R}^m\) and \(\vec A\equiv(A_1, A_2)\). Denote by \({\mathcal A}_\infty(\rnm; \vec A)\) the class of Muckenhoupt weights associated with \(\vec A\). The authors introduce a class of anisotropic singular integrals on \(\mathbb R^n\times\mathbb R^m\), whose kernels are adapted to \(\vec A\) in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on \(L^q_w(\mathbb R^n\times\mathbb R^m)\) with \(q\in(1, \infty)\) and \(w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)\) or on \(H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)\) with \(p\in(0, 1]\) and \(w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)\). These results are also new even when \(w=1\). |
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| AbstractList | Let \(A_i\) for \(i=1, 2\) be an expansive dilation, respectively, on \({\mathbb R}^n\) and \({\mathbb R}^m\) and \(\vec A\equiv(A_1, A_2)\). Denote by \({\mathcal A}_\infty(\rnm; \vec A)\) the class of Muckenhoupt weights associated with \(\vec A\). The authors introduce a class of anisotropic singular integrals on \(\mathbb R^n\times\mathbb R^m\), whose kernels are adapted to \(\vec A\) in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on \(L^q_w(\mathbb R^n\times\mathbb R^m)\) with \(q\in(1, \infty)\) and \(w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)\) or on \(H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)\) with \(p\in(0, 1]\) and \(w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)\). These results are also new even when \(w=1\). |
| Author | Bownik, Marcin Li, Baode Yang, Dachun Zhou, Yuan |
| Author_xml | – sequence: 1 givenname: Baode surname: Li fullname: Li, Baode – sequence: 2 givenname: Marcin surname: Bownik fullname: Bownik, Marcin – sequence: 3 givenname: Dachun surname: Yang fullname: Yang, Dachun – sequence: 4 givenname: Yuan surname: Zhou fullname: Zhou, Yuan |
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| Copyright | 2010. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| DOI | 10.48550/arxiv.0903.4720 |
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| Snippet | Let \(A_i\) for \(i=1, 2\) be an expansive dilation, respectively, on \({\mathbb R}^n\) and \({\mathbb R}^m\) and \(\vec A\equiv(A_1, A_2)\). Denote by... |
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| Title | Anisotropic Singular Integrals in Product Spaces |
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