scholarly journals Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets

2019 ◽  
Vol 12 (7) ◽  
pp. 1843-1890
Author(s):  
Svitlana Mayboroda ◽  
Zihui Zhao
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Measure ◽  
Absolute Continuity ◽  
Square Function ◽  
Square Function Estimates ◽  
Lower Dimensional

scholarly journals Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Mingming Cao ◽  
José María Martell ◽  
Andrea Olivo
Keyword(s):  
Absolute Continuity ◽  
Square Function ◽  
Surface Measure ◽  
Antisymmetric Part ◽  
Null Solution ◽  
Almost Everywhere ◽  
Square Function Estimates ◽  
The Absolute ◽  
The Laplace Operator ◽  
Conical Square Function

AbstractIn nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators $$L_1$$ L 1 and $$L_2$$ L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of $$L_1$$ L 1 is equivalent to the same property for $$L_2$$ L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which $$L_2$$ L 2 is either the transpose of $$L_1$$ L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.

scholarly journals Absolute continuity of parabolic measure and the initial-dirichlet problem

2019 ◽  
Author(s):  
◽  
Alyssa Genschaw
Keyword(s):  
Dirichlet Problem ◽  
Absolute Continuity ◽  
Divergence Form ◽  
Surface Measure ◽  
Scale Invariant ◽  
Elliptic Case ◽  
Open Set ◽  
Caloric Measure ◽  
Reverse Hölder ◽  
Hölder Estimate

This thesis is devoted to the study of parabolic measure corresponding to a divergence form parabolic operator. We first extend to the parabolic setting a number of basic results that are well known in the elliptic case. Then following a result of Bennewitz-Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set [omega] R(n+1), assuming as a background hypothesis only that the essential boundary of [omega] satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We then show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with "lateral" data in [Lp], for some p less than [infinity]. Finally, we prove that for the heat equation, BMO-solvability implies scale invariant quantitative absolute continuity of caloric measure with respect to surface measure, in an open set [omega] with time-backwards ADR boundary. Moreover, the same results apply to the parabolic measure associated to a uniformly parabolic divergence form operator (L), with estimates depending only on dimension, the ADR constants, and parabolicity, provided that the continuous Dirichlet problem is solvable for (L) in [omega]. By a result of Fabes, Garofalo and Lanconelli [FGL], this includes the case of [C1]-Dini coefficients.

scholarly journals Stability estimates for discrete harmonic functions on product domains

2013 ◽  
Vol 7 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Maru Guadie
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Harmonic Measure ◽  
Harmonic Functions ◽  
Discrete Analog ◽  
Stability Estimates ◽  
Product Domains ◽  
Discrete Harmonic Functions

We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm?n-Lindel?f theorem and use Fourier series to obtain a discrete analog of the three-line theorem for the gradients of harmonic functions in a strip. Then we derive estimates for the discrete harmonic measure and use elementary spectral inequalities to obtain stability estimates for Dirichlet problem in cylinder domains.

scholarly journals Absolute continuity of harmonic measure for domains with lower regular boundaries

2019 ◽  
Vol 345 ◽  
pp. 1206-1252 ◽  
Author(s):  
Murat Akman ◽  
Jonas Azzam ◽  
Mihalis Mourgoglou
Keyword(s):  
Harmonic Measure ◽  
Absolute Continuity

scholarly journals BMO Solvability and Absolute Continuity of Harmonic Measure

2017 ◽  
Vol 28 (4) ◽  
pp. 3278-3299 ◽  
Author(s):  
Steve Hofmann ◽  
Phi Le
Keyword(s):  
Harmonic Measure ◽  
Absolute Continuity
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