Sharp maximal function inequalities and boundedness for Toeplitz type operator related to general fractional integral operators
The author considers some operators modelled by Toeplitz operators and integral operators with general kernels and proves some inequalities concerning estimates between maximal functions associated to them. Namely, for a kernel family \(F_t(x,y)\), \(t\geq 0\), \((x,y)\in \mathbb R^n \times \mathbb R^n\), a given Banach space \(H\) (of functions defined on \([0,\infty)\)), he deals with the operators \(F_t(f)(x)=\int_{\mathbb R^n} F_t(x,y)f(y)\,dy\) and \(T(f)(x)= \|F_t(f)(x)\|_H\), where the norm is taken in the \(t\)-variable and \(T\) is assumed to be bounded on \(L^2(\mathbb R^n)\). Given a locally integrable function \(b\) and inspired by the commutator \([b,T]=bT(f)-T(bf)\), he considers combinations \(F^b_t(f)(x)=\sum_{k=1}^m F_t^{k,1}(M_b I_\alpha F^{k,2}_t (f))+ F_t^{k,3}(I_\alpha M_b F^{k,4}_t (f)) \), where \(F^{k,1}(f)\) stands for \(F_t(f)\) or \(\pm I\), \(\|F_t^{k,2}(f)\|_H\), \(\|F_t^{k,3}(f)\|_H\) and \(\|F_t^{k,4}(f)\|_H\) are bounded operators on \(L^p(\mathbb R^n)\), \(I_\alpha\) is the fractional integral and \(M_b\) is the multiplication by \(b\). Denoting \(T^b(f)= \|F^b_t (f)(x)\|\), it is shown, under certain conditions on the kernel functions \(F_t\) (Calderón-Zygmund type conditions) and \(b\in \mathrm{Lip}_\beta\), the boundedness of \(T\) and \(T^b\) in spaces such as \(L^p\) or generalized Morrey spaces. The author presents some applications to Littlewood-Paley, Marcinkiewicz and maximal Bochner-Riesz operators, where the space \(H\) is selected in a convenient way in each case. Reviewer's remark. Very similar results with the same introduction, analogous proof and applications have been given in [\textit{S. Guo}, \textit{C.-X. Huang} and \textit{L.-Z. Liu}, An. Univ. Craiova, Ser. Mat. Inf. 39, No. 1, 35--47 (2012; Zbl 1274.42033), \url{http://inf.ucv.ro/~ami/index.php/ami/article/view/442/402}], with the main difference that the fractional integral operator is not considered there.
scientific article; zbMATH DE number 6147126