Weighted estimates of a measure of noncompactness for maximal and potential operators
The authors are concerned with estimates for the norm on space \(L^p_w(G)\), where \(G\) stands for a homogeneous group and \(w\) represents the weight function in defining the norm by usual formula. They are particularly interested in the ``maximal functions'' defined by \[ M_\alpha f(x)= \sup_{B\ni x}\,{1\over|B|^{1-\alpha/Q}}\int_B |f(y)|\,dy,\quad x\in G,\quad 0\leq\alpha< Q, \] and Riesz potentials of the form \[ I_\alpha f(x)= \int_G {f(y)\over r(xy^{-1})^{Q-\alpha}}\,dy,\quad 0<\alpha< Q. \] Several estimates are obtained for such integrals, mainly lower estimates, and the following noncompactness result is obtained: Theorem. Let \(p\in (1,\infty)\) and assume that the operator \(M_0\) is bounded from \(L^p_w(G)\) into \(L^p_v(G)\). Then there is no pair \((w,v)\) such that \(M_0\) is also compact. Applications are given, including one to the operator of partial sums for Fourier series.
scientific article; zbMATH DE number 5352056