Weighted inequalities for rough square functions through extrapolation (Q2773421)

The study of \(L^p\) inequalities for some operators defined through NEWLINE\[NEWLINEg(f)(x)= \Biggl(\int^\infty_0|N_t* f(x)|^2{dt\over t}\Biggr)^{1/2},\tag{1}NEWLINE\]NEWLINE where \(N_t\), \(t > 0\), are dilatations of a certain fixed function \(N\), has been carried out by different authors. That is the case of Littlewood-Paley and Marcinkiewicz type operators [\textit{E. M. Stein}, Trans. Am. Math. Soc. 88, 430-466 (1958; Zbl 0105.05104) and \textit{A. Benedek}, \textit{A. P. Calderón} and \textit{R. Panzone}, Proc. Natl. Acad. Sci. USA 48, 356-365 (1962; Zbl 0103.33402)].NEWLINENEWLINENEWLINEResults on \(L^p(w)\) spaces with \(w\) in the Muckenhoupt class of weights \(A_p\) have been also obtained for the Littlewood-Paley and Marcinkiewicz operators ([\textit{D. S. Kurtz}, Trans. Am. Math. Soc. 259, 235-254 (1980; Zbl 0436.42012)] and [\textit{A. Torchinsky} and \textit{S. Wang}, Colloq. Math. 60/61, No. 1, 235-243 (1990; Zbl 0731.42019)]) and for square functions with rough kernels [\textit{Y. Ding}, \textit{D. Fan} and \textit{Y. Pan}, Indiana Univ. Math. J. 48, No. 3, 1037-1055 (1999; Zbl 0949.42014)].NEWLINENEWLINENEWLINEIn this paper the authors consider first the square function where \(N_t\), \(t> 0\), are not took in general as dilatations of a function \(N\). They prove that the square function is bounded on \(L^2(w^s)\), \(0\leq s< 1\), if \(w\in A_2\) is a uniform weight for the operators given by convolution with \(N_t\) and if some decay properties for the Fourier transforms of \(N_t\) are satisfied.NEWLINENEWLINENEWLINEBy considering the extrapolation theorem of \textit{J. L. Rubio de Francia} [Am. J. Math. 106, 533-547 (1984; Zbl 0558.42012)] the authors extend the results to weighted \(L^p\) spaces, for weights in the Muckenhoupt class. Moreover, the extrapolation argument allows them to obtain weighted \(L^p\) inequalities for a larger class of weights related to a maximal function.NEWLINENEWLINENEWLINEAs applications of these results they prove the boundedness of square functions with rough kernels on \(L^p(w)\) spaces which extend and improve some previous results [\textit{S. Sato}, Bull. Aust. Math. Soc. 58, No. 2, 199-211 (1998; Zbl 0914.42012) and \textit{Y. Ding}, \textit{D. Fan} and \textit{Y. Pan} (loc. cit.)].NEWLINENEWLINENEWLINETheir method is also useful in the study of weighted inequalities for square functions that are defined with a family \(N_t\) of singular finite Borel measures and where the weights are related to the spherical maximal function.NEWLINENEWLINENEWLINEIn the paper it is also indicated that similar theorems can be adapted with the appropriate modifications to product spaces and to the context of spaces of homogeneous type.