Search

Abstract: In this paper we study the centered and non-centered Hardy–Littlewood maximal functions, M and , for a wide class of non-doubling measure μ defined on cusped manifolds. In particular, we give a class of manifolds with exponential volume growth on which M is of weak type and is bounded on for all . [Copyright &y& Elsevier]

Let Ω⊂RN be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the L2-norm of U term is removed, where K is a closed subset of Ω and U is a function on Ω&z.drule;K with gradient in L2. The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the HN−1-measure of K∩B(x,r) is bounded above by crN−1 and bounded below by c−1rN−1. [Copyright &y& Elsevier]