Square Function Estimates and the T(b) Theorem

1990 ◽  
Vol 110 (3) ◽  
pp. 721
Author(s):  
Stephen Semmes
Keyword(s):  
Square Function ◽  
Square Function Estimates

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 Banach space operators with a bounded H∞ functional calculus

1996 ◽  
Vol 60 (1) ◽  
pp. 51-89 ◽  
Author(s):  
Michael Cowling ◽  
Ian Doust ◽  
Alan Micintosh ◽  
Atsushi Yagi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Functional Calculus ◽  
General Definition ◽  
Square Function ◽  
Square Function Estimates ◽  
Imaginary Powers ◽  
Banach Space Operators ◽  
General Banach Space

AbstractIn this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when f is holomorphic on a larger sector.We also examine how certain properties of this functional calculus, such as the existence of a bounded H∈ functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive Lp space, then T has a bounded H∈ functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.

The H∞-Functional Calculus and Square Function Estimates

2015 ◽  
pp. 715-771 ◽  
Author(s):  
Fritz Gesztesy ◽  
Gilles Godefroy ◽  
Loukas Grafakos ◽  
Igor Verbitsky
Keyword(s):  
Functional Calculus ◽  
Square Function ◽  
Square Function Estimates

scholarly journals On the failure of lower square function estimates in the non-homogeneous weighted setting

2018 ◽  
Vol 374 (3-4) ◽  
pp. 1923-1952 ◽  
Author(s):  
K. Domelevo ◽  
P. Ivanisvili ◽  
S. Petermichl ◽  
S. Treil ◽  
A. Volberg
Keyword(s):  
Square Function ◽  
Square Function Estimates
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