Sunrise strategy for the continuity of maximal operators

Abstract

In this paper we address the W1,1-continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into “lateral” maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy—Littlewood maximal operator $$\tilde M$$ acting on W rad 1,1 (ℝd), the subspace of W1,1(ℝd) consisting of radial functions. In dimension d ≥ 2 it was recently established by H. Luiro that the map $$f \mapsto \nabla \tilde Mf$$ is bounded from W rad 1,1 (ℝd) to L1(ℝd), and we show that such a map is also continuous. Further applications of the sunrise strategy in connection with the W1,1-continuity problem include non-tangential maximal operators on ℝd acting on radial functions when d ≥ 2 and general functions when d = 1, and the uncentered Hardy—Littlewood maximal operator on the sphere $${^d}$$ acting on polar functions when d ≥ 2 and general functions when d = 1.