Discrete harmonic measure, Green's functions and symmetrization: A unified probabilistic approach (Q2769241)

Let \((X,\mathcal F, \mu)\) be a measure space. In this paper a general symmetrisation procedure for subsets of \(X\) and \(X\times X\) is discussed which includes (and generalises) the usual Steiner symmetrisation. Various integral inequalities for symmetrised functions are derived. The main focus of the paper are inequalities for (discrete) harmonic measures and Green functions involving symmetrizations. In particular, new versions of Baernstein's theorem on the effects of circular symmetrizations on harmonic measure and Green's function are obtained and a discrete version (on \(\mathbb Z\times \mathbb Z_m\)) of Beurling's shove theorem is proved.