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On the boundedness of a class of rough maximal operators on product spaces. (English) Zbl 1220.42011
The authors study the $$L^p$$ boundedness of maximal operators related with classes of singular integrals on product spaces. The main result is the following:
Suppose that $$\{\Omega_j\}$$ is a fixed countable subset of $$L^q (\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})$$ for some $$1<q\leq \infty$$ with $$\| \|\Omega_j\|_{L^q (\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})} \|_{l^{\gamma'}}<\infty$$. Assume that $$\Phi$$ and $$\Psi\in C^2 ([0,\infty))$$ are convex and increasing functions with $$\Phi(0)=\Psi(0)=0.$$ Let $$E^{(\gamma)}(\{\Omega_j\})$$, $$1\leq \gamma <\infty$$, denote the class of all kernels of the form
$K(u,v)=\sum_{j} h_j (|u|,|v|)\;\frac{\Omega_j(u,v)}{|u|^{n}|v|^{m}},$
where
$\bigg(\int_{0}^{\infty} \int_{0}^{\infty} \sum_{j} |h_j(r,t)|^\gamma \,\frac{dr\,dt}{rt}\bigg)^{1/\gamma}\leq 1;$
and let
\begin{aligned} T_{K,\Phi,\Psi}f(x,y)&= \text{p.v. }\int_{\mathbb{R}^n\times \mathbb{R}^m}f\bigg(x-\Phi(|u|)\frac{u}{\|u\|},y-\Psi(|v|)\frac{v}{\|v\|}\bigg)K(u,v)\,du\, dv,\\ T_{\Phi,\Psi,\{\Omega_j\}}^{(\gamma)}(f)&= \sup \big\{|T_{K,\Phi,\Psi}f|:K\in E^{(\gamma)}(\{\Omega_j\}) \big\}. \end{aligned}
Then the inequality
$\|T_{\Phi,\Psi,\{\Omega_j\}}^{(\gamma)}(f)\|_{L^p (\mathbb{R}^{n}\times \mathbb{R}^{m})} \leq C_p \bigg(\frac{q}{q-1}\bigg)^{2/\gamma'}\| \|\Omega_j\|_{L^q (\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})} \|_{l^{\gamma'}}\|f\|_{L^p (\mathbb{R}^{n}\times \mathbb{R}^{m})}$
holds for $$(\alpha\beta\gamma')/(\gamma'\alpha+\alpha\beta-\gamma')<p<\infty$$ and $$1\leq \gamma \leq 2$$, where $$\alpha=\min(m,n)$$ and $$\beta=\max(2,q')$$.
By this theorem and by applying an extrapolation method, some new and improved results for maximal operators and singular integrals on product spaces are obtained.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory
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