Weighted bounds for the Carleson maximal operator in \(\mathbb{R}^ n\)
Consider the Fourier multiplier (spherical partial sum operator) \(S_ R\) given by \((S_ R f)\sphat (\xi)= \chi_{\{|\xi|< R\}}\widehat f(\xi)\), and define the maximal operator \(T^*\) (Carleson maximal operator) by \(T^* f(x)= \sup_{R> 0}| S_ R f(x)|\). The author shows the following: Let \(1< p< \infty\) and let \(w(r)\) be a positive function with \(w(r) r^{(n-1)(1- p/2)}\in A_ p(\mathbb{R}^ +)\). Then there exists a constant \(C\) such that \[ \int_{\mathbb{R}^ n} | T^*(fP)(x)|^ p w(| x|) dx\leq C\int_{\mathbb{R}^ n} | f_ 0(| x|) P(x)|^ p w(| x|) dx \] for every radial function \(f(x)= f_ 0(| x|)\) and every homogeneous harmonic polynomial \(P(x)\). Here \(A_ p(\mathbb{R}^ +)\) coincides with the set of all even functions on \(\mathbb{R}\) which belong to the Muckenhoupt \(A_ p\) class . Unweighted and radial function case was treated by the authoress and \textit{F. Soria} in Proc. Am. Math. Soc. 111, No. 4, 1015- 1022 (1991; Zbl 0723.42004).
scientific article; zbMATH DE number 722326
scientific article