BMO solvability and the $A_∞$ condition for second order parabolic operators

Abstract

General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately and investigate your claim.

The Dirichlet problem for second order parabolic operators in divergence form

“…In particular, our main result, which is new already in the case when A is symmetric and time-dependent, gives a parabolic analogue of the main result in [14] concerning elliptic measure. Our proof heavily relies on square function estimates and non-tangential estimates for parabolic operators with time-dependent coefficients that were only recently obtained by us in [4] as well as the reduction to a Carleson measure estimate proved in [9]. As we shall avoid the change of variables utilized in [14], this also gives a simpler and more direct proof of the A ∞ -property of elliptic measure.…”

“…Remark 1.7. Theorem 1.6 is a priori equivalent to the statement that (1.5) holds for all parabolic cubes whenever u is the unique solution to the continuous Dirichlet problem for Hu = 0 with continuous compactly supported boundary data f satisfying |f | ≤ 1, see Remark 5 in [9]. Note that in this case |u| ≤ 1 by the maximum principle.…”

“…The proof consists of three parts: a reduction to a Carleson measure estimate, the construction of a particular set F , and the proof of the Carleson measure estimate by partial integration. These three parts have four sources of insights [4,9,14,20]. In general, c will denote a generic constant, not necessarily the same at each instance, which, unless otherwise stated, only depends on n and the ellipticity constants.…”

Parabolic Regularity and Dirichlet boundary value problems

“…In particular in this paper the authors has solved the L 2 Dirichlet problem for the heat equation in graph domains of Lewis-Murray type. A related class of localised domains in which parabolic boundary value problems are solvable was considered in [Riv14] as well as in [DH16,DPP16]. The paper [DH16] has established L p solvability for parabolic Dirichlet problem under assumption that the coefficients satisfy certain natural small Carleson condition which also appears for elliptic PDEs.…”

“…The paper [DH16] has established L p solvability for parabolic Dirichlet problem under assumption that the coefficients satisfy certain natural small Carleson condition which also appears for elliptic PDEs. The second paper [DPP16] finds sufficient and necessary condition for the parabolic measure to be A ∞ with respect to the parabolic analogue of the surface measure.…”