Marcinkiewicz integral operators along twisted surfaces
In this paper, the author defined a class of Marcinkiewicz integral operators \(\mathcal{M}_{\Gamma_{P,Q},\Omega,h}\) on product domains by translates determined by twisted surfaces. More precisely, \(\mathcal{M}_{\Gamma_{P,Q},\Omega,h}\) is defined by \[\mathcal{M}_{\Gamma_{P,Q},\Omega,h}f(x,y)=\bigg(\iint_{\mathbb{R}\times\mathbb{R}}|F_{t,s,\Gamma_\alpha,\Omega,h}f(x,y)|^2\frac{dtds}{2^{2t}2^{2s}}\bigg)^{1/2},\] where \[F_{t,s,\Gamma_{P,Q},\Omega,h}f(x,y)=\int_{|u|\le2^t}\int_{|v|\le2^s}f((x,y)-\Gamma_{P,Q}(u,v))\frac {\Omega(u,v)h(|u|,|v|)}{u^{n-1}|v|^{m-1}}dudv.\] Further, \(h\in L^\infty(\mathbb{R}_+\times\mathbb{R}_+)\), \(\Gamma_{P,Q}=(P(|y|)x,Q(|x|)y)\) with \(P,Q:\mathbb{R}\rightarrow\mathbb{R}\) are polynomials and \(\Omega\in L^1(\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})\) satisfying: \[\Omega(tx,sy)=\Omega(x,y)\ \text {for any }\ t,s>0,\] \[\int_{\mathbb{S}^{n-1}}\Omega(u^\prime,\cdot)d\sigma_n(u^\prime)=\int_{\mathbb{S}^{m-1}}\Omega(\cdot,v^\prime)d\sigma_m(v^\prime)=0.\] With the above definitions, in this paper, the author gives the \(L^p(\mathbb{R}^n\times\mathbb{R}^m)\) boundedness of the Marcinkiewicz integral operator \(\mathcal{M}_{\Gamma_{P,Q},\Omega,h}\) when \(\Omega\in L(\log^+L)(\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})\) and the polynomials \(P,Q\) satisfy some conditions. To prove the above result, the author introduces a two-parameter maximal operator and gives also the \(L^p(\mathbb{R}^n\times\mathbb{R}^m)\) boundedness of this maximal operator.
scientific article; zbMATH DE number 7461704
scientific article