Anisotropic Singular Integrals in Product Spaces - Li, Baode; Bownik, Marcin; Yang, Dachun; Zhou, Yuan (arXiv.org, 2010)
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 \({\mathcal A}_∞(\rnm; \vec A)\) the class of Muckenhoupt weights associated with \(\vec A\). The authors introduce a class of anisotropic singular integrals on \(\mathbb R^n×\mathbb R^m\), 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 \(L^q_w(\mathbb R^n×\mathbb R^m)\) with \(q\in(1, ∞)\) and \(w\in\mathcal A_q(\mathbb R^n×\mathbb R^m; \vec A)\) or on \(H^p_w(\mathbb R^n×\mathbb R^m; \vec A)\) with \(p\in(0, 1]\) and \(w\in\mathcal A_∞(\mathbb R^n ×\mathbb R^m; \vec A)\). These results are also new even when \(w=1\).