Maximal operators and singular integrals with non-isotropic dilation on product domains
Let \(n_{1},n_{2}\geq 2\) and let \(P_{j}\) be a real \(n_{j}\times n_{j}\) matrix whose eigenvalues have positive real parts. Let \(\gamma _{j}\) \(=\) trace \(P_{j}\) for \(j\in \{1,2\}.\) For \(j\in \{1,2\},\) define a dilation group \(\{A_{j,t}\}_{t>0}\) on \(\mathbb{R}^{n_{j}}\) by \(A_{j,t}=t^{P_{j}}=\exp (P_{j}\log t).\) Then there exists a non-negative function \(\rho _{j}\) on \( \mathbb{R}^{n_{j}}\) associated with \(\{A_{j,t}\}_{t>0}\) such that \(\rho _{j}\) is continuous on \(\mathbb{R}^{n_{j}}\) and infinitely differentiable \ on \( \mathbb{R}^{n_{j}}\backslash \{0\}\) and satisfies the following: (i) \(\rho _{j}(A_{j,t}x_{j})=t\rho _{j}(x_{j})\) for all positive values of \(t\) and \( x_{j}\in \mathbb{R}^{n_{j}};\) (ii) \(\rho _{j}(x_{j}+y_{j})\leq C\left( \rho _{j}(x_{j})+\rho _{j}(y_{j})\right) \) for all \(x_{j},y_{j}\in \mathbb{R} ^{n_{j}}\) and for some positive constant \(C;\) (iii) If \(\sum_{j}=\left\{ x_{j}:\rho _{j}(x_{j})=1\right\} ,\) then \(\sum_{j}=\left\{ \theta _{j}:\left\langle B_{j}\theta _{j},\theta _{j}\right\rangle )=1\right\} \) for a positive symmetric matrix \(B_{j}\). Then the Lebesgue measure \( dx_{j}=t^{^{\gamma _{j}}-1}d\sigma _{j}dt.\) That is, \[ \int_{\mathbb{R}^{n_{j}}}f(x_{j})dx_{j}=\int_{0}^{\infty }\int_{\sum_{j}}f(A_{j,t}\theta _{j})d\sigma _{j}(\theta _{j})dt \] for appropriate functions \(f,\) where \(d\sigma _{j}\) is a \(C^{\infty }\) \ measure on \(\sum_{j},\) \(j\in \{1,2\}\). Let \(\Omega \in L^{1}(\sum_{1}\times \sum_{2})\) and let \(\Omega\) satisfy \(\Omega (A_{1,t_{1}}x_{1},A_{2,t_{2}}x_{2})=\Omega (x_{1},x_{2})\) for all \( t_{1},t_{2}>0\) and \[ \int_{\sum_{1}}\Omega (\theta _{1},\theta _{2})d\sigma _{1}(\theta _{1})=\int_{\sum_{2}}\Omega (\theta _{1},\theta _{2})d\sigma _{2}(\theta _{2})=0. \] Define the non-isotropic singular integral operator \(T_{\Omega ,h}\) on \( \mathbb{R}^{n_{1}}\times \mathbb{R}^{n_{2}}\) by \[ T_{\Omega ,h}f(x_{1},x_{2})=\text{p.v.}\int \int_{\mathbb{R}^{n_{1}}\times \mathbb{R}^{n_{2}}}f(x_{1}-y_{1},x_{2}-y_{2})K_{\Omega ,h}(y_{1},y_{2})dy_{1}dy_{2}, \] and its related maximal operator \[ M_{\Omega }^{(\nu )}f(x_{1},x_{2})=\sup_{h\in U_{\nu }}\left| T_{\Omega ,h}f(x_{1},x_{2})\right| , \] where \(K_{\Omega ,h}(y_{1},y_{2})=\Omega (y_{1}^{\prime },y_{2}^{\prime })\rho _{1}(y_{1})^{-\gamma _{1}}\rho _{2}(y_{2})^{-\gamma _{2}}h(\rho _{1}(y_{1}),\rho _{2}(y_{2})),y_{j}^{\prime }=A_{j,\rho _{j}^{-1}(y_{j})}y_{j}\) and \(U_{\nu }\) is the set of all \(h\) satisfying \( \left\| h\right\| _{L^{\nu }(\mathbb{R}_{+}\times \mathbb{R}_{+},\frac{ dr_{1}dr_{2}}{r_{1}r_{2}})}=\left( \int \int_{\mathbb{R}_{+}\times \mathbb{R} _{+}}\left| h(r_{1},r_{2})\right| ^{\nu }\frac{dr_{1}dr_{2}}{ r_{1}r_{2}}\right) ^{1/\nu }\leq 1.\) The main results of the paper under review are the following: Theorem 1. Let \(1\leq \nu \leq 2\). If \(\Omega \in L\left( \log L\right) ^{2/\nu ^{\prime }}(\sum_{1}\times \sum_{2}),\) then \(M_{\Omega }^{(\nu )}\) is bounded on \(L^{p}(\mathbb{R}^{n_{1}}\times \mathbb{R}^{n_{2}})\) for \(\nu ^{\prime }\leq p<\infty \) and \(1<\nu \leq 2\) and it is bounded on \(L^{\infty }(\mathbb{R}^{n_{1}}\times \mathbb{R}^{n_{2}})\) if \(\nu =1\) and \(\Omega \in L^{1}(\sum_{1}\times \sum_{2}).\) Theorem 2. If \(\Omega \in L\left( \log L\right) ^{2/\nu ^{\prime }}(\sum_{1}\times \sum_{2})\) and \(h\in L^{\nu }(\mathbb{R}_{+}\times \mathbb{ R}_{+},\frac{dr_{1}dr_{2}}{r_{1}r_{2}})\) for \(1<\nu \leq \infty ,\) then \( T_{\Omega ,h}\) is bounded on \(L^{p}(\mathbb{R}^{n_{1}}\times \mathbb{R} ^{n_{2}})\) for \(1<p<\infty .\) We remark that \ for the special cases \(\rho _{1}(y_{1})=\left| y_{1}\right| \) and \(\rho _{2}(y_{2})=\left| y_{2}\right| \) (the Euclidean norm), the above results were obtained for \(M_{\Omega }^{(\nu )}\) (in the case \(\nu =1\) and only for \(p=2)\) by \textit{Y. Ding} [J. Math. Anal. Appl. 232, No.1, 222--228 (1999; Zbl 0920.42013)], for both \(M_{\Omega }^{(\nu )}\) and \( T_{\Omega ,h}\) (in the case \(\nu =2\) and \(p\) as above\()\) by \textit{A. Al-Salman} [J. Math. Anal. Appl. 311, No. 1, 338--351 (2005; Zbl 1085.42009)] and for both \( M_{\Omega }^{(\nu )}\) and \(T_{\Omega ,h}\) (in the case \(\nu \) and \(p\) as Theorems 1 and 2\()\) by \textit{H. Al-Qassem} and \textit{Y. Pan} [J. Integral Equations Appl. 17, No. 4, 331--356 (2005; Zbl 1130.42015)]. Also, the optimality of the condition \(\Omega \in L\left( \log L\right) ^{2/\nu ^{\prime }}(\sum_{1}\times \sum_{2})\) in the case \(\nu =2\) was independently proved by \textit{A. Al-Salman} (loc. cit.) and by \textit{H. Al-Qassem} and \textit{Y. Pan} (loc. cit.).
scientific article; zbMATH DE number 5808617
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