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Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

Simon Bortz, John Hoffman, Steve Hofmann, José Luis Luna García and Kaj Nyström

Vol. 16 (2023), No. 4, 1061–1088

Let E n+1 be a parabolic uniformly rectifiable set. We prove that every bounded solution u to

tu Δu = 0 in  n+1 E

satisfies a Carleson measure estimate condition. An important technical novelty of our work is that we develop a corona domain approximation scheme for E in terms of regular Lip (12,1) graph domains. This scheme has an analogous elliptic version which improves on the known results in that setting.

parabolic equations, parabolic uniform rectifiability, Carleson measures
Mathematical Subject Classification
Primary: 28A75
Secondary: 28A78, 35K10, 42B37
Received: 29 March 2021
Revised: 23 September 2021
Accepted: 26 October 2021
Published: 15 June 2023
Simon Bortz
Department of Mathematics
University of Alabama
Tuscaloosa, AL
United States
John Hoffman
Department of Mathematics
University of Missouri at Columbia
Columbia, MO
United States
Steve Hofmann
Department of Mathematics
University of Missouri at Columbia
Columbia, MO
United States
José Luis Luna García
Department of Mathematics & Statistics
McMaster University
Kaj Nyström
Department of Mathematics
Uppsala University

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