Caltech/UCLA Joint Analysis Seminar

SEMINAR WILL BE HELD AT UCLA MS 6627

Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weak-type (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than Llogloglog L. This is based on joint work with Ben Krause.