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Carleson measure estimates for bounded harmonic functions

John Garnett, UCLA
Tuesday, October 26, 2021 - 1:30pm to 3:30pm

Let $\Omega$ be a domain in $R^{d+1}$ where $d \geq 1$.  It is known that (using definitions  given  at the start of the talk) if $\Omega$ satisfies a corkscrew condition and  $\partial \Omega$ is $d$-Ahlfors, then the following are equivalent:

(a)   a square function Carleson measure estimate holds for all bounded harmonic functions on $\Omega;$

(b) an $\varepsilon$-approximation property holds for all such functions and all $0 < \varepsilon < 1;$

(c) $\partial \Omega$ is uniformly rectifiable.

 Here we explore (a) and (b) when $\partial \Omega$  is not required to be Ahlfors regular.

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