Joint BCAM-UPV/EHU Analysis and PDE seminar: Caloric measure and regular Lip(1,1/2) graphs

Abstract

Let Ω be the domain above the graph Σ of a Lip(1,1/2) function a(x, t), defined on Rn - R. We endow the boundary Σ with the natural "Surface measure" σ := Hpn+1b Σ, where Hn+1p is the parabolic Hausdorff measure of homogeneous dimension n + 1. About 30 years ago, J. L. Lewis and M. A. M. Murray proved that the Dirichlet problem is solvable in Ω, with data in Lp (Σ, dσ) for some p < ∞, provided that, in addition, a(x, t) is a regular Lip(1,1/2) function, i.e., that a certain half-order time derivative of a lies in parabolic BMO(Rn). It is well known that, in this context, the Lp solvability result is equivalent to a certain scale invariant absolute continuity property of caloric measure ω with respect to σ, namely, that ω ∈ A∞(σ), where A∞ is the usual Muckenhoupt class. On the other hand, a classical example of Kaufman and Wu shows that absolute continuity of ω with respect to σ fails in general, for Ω as above with a Lip(1,1/2) boundary. The question had remained open whether the regular Lip(1,1/2) condition was necessary as well as sufficient. In this talk we shall discuss a proof of the fact that the A∞ property does indeed imply regularity of Σ; thus we establish a converse to the sufficiency result of Lewis and Murray.
The method of proof is based on establishing Littlewood-Paley type estimates for the level sets of the caloric Green function.
We remark that for a Lip(1,1/2) graph Σ, being regular is equivalent to being uniformly rectifiable in the parabolic sense. It remains an open problem to show that, more generally, for an open set in parabolic space-time Rn+1 with a parabolic Ahlfors regular boundary, the (weak)-A∞ property of ω implies (parabolic) uniform rectifiability of the boundary. If time permits, we´ll discuss some (possible?) progress towards this goal, and explain the two fundamental obstacles that arise in the parabolic context, but not
in the elliptic setting (where the analogous result is now known).

This is joint work with S. Bortz, J. M. Martell and K. Nystr ̈om.



Link to the session: 
https://us06web.zoom.us/j/99649860282?pwd=SE0vemtYMFlwbFBNTXQyOTBONG0vZz09

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