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Weighted estimates of a measure of noncompactness for maximal and potential operators. (English) Zbl 1153.47044
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.
MSC:
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H50 Potential operators (MSC2000)
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