TWO RESULTS OVER SETS WITH BIG PIECES OF PARABOLIC LIPSCHITZ GRAPHS

Abstract. For a set E in (n + 1)-Euclidean space with uniform big pieces of parabolic Lipschitz graphs (as defined in the bulk of the paper) we first observe that certain parabolic singular integrals are bounded on E. If E is the boundary of certain type of non-cylindrical domain, and E is regular for the heat equation, we prove a weak reverse Hölder inequality for caloric measure over E. 1. Definitions Our ultimate goal in this note is to obtain a result on singular integrals of parabolic type, and a result on caloric measure, on sets which are approximable by “good graphs”. In certain way we follow the structure and motivation of [3], and so we first provide the definitions used throughout this work. For (X, t) ∈ Rn × R ≡ Rn+1, denote by Cr(X, t) the cylinder {(Y, s) ∈ Rn+1: |X − Y | < r, |t − s | < r2}, and let ∆r(X, t) denote the surface cube Cr(X, t) ∩ ∂Ω. The parabolic cylinder of radius r> 0 and centered at (X, t) is defined as Cr(X, t) = {(Y, s) ∈ Rn+1: |X − Y | < r, 0 < t − s < r2}. The parabolic ball of radius r> 0 centered at (X, t) is Qr(X, t) = {(Y, s) ∈ Rn+1: |X−Y |+ |t−s|1/2 < r}. The parabolic distance between (X, t), (Y, s) ∈ Rn+1 is defined by d(X, t;Y, s) = |X − Y | + |t − s|1/2 ≡ ‖X − Y, t − s‖. This last expression defines what we will call the parabolic norm of points in Rn+1, and it may also be applied to points 2000 Mathematics Subject Classification. 42B20, 35K20. Key words and phrases. Parabolic uniform rectifible sets, big pieces of parabolic Lipschitz graphs, parabolic singular integrals, caloric measure, weak reverse Hölder inequalities