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Uniqueness for Spherically Convergent Multiple Trigonometric Series
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
A major tool for the remaining argument is harmonic measure. Fix an open set G ⊂ 𝕋2 and fix a point x ∈ G. Then the harmonic measure ω(G, ·, x) is a probability measure on ∂G, the boundary of G. If F is a subset of ∂G, then its harmonic measure ω(G, F, x) is the probability that a Brownian motion starting at x will first intersect the boundary of G at some point of F. The reason for the name is that if we switch perspectives by fixing F and G and then letting x vary, the resulting function of x is harmonic. In fact it can be thought of as the solution to Dirichlet’s problem, the problem of finding a harmonic function in G which tends to 1 on any sequence of points tending to a point of F and which tends to 0 on any sequence of points tending to a point of ∂G \ F. It is this sort of frequent change of point of view that motives using the democratic notation that we do, instead of something like ωG,x(F). By the maximum principle, for every x ∈ G, 0 < ω(G, F, x) < 1, unless F = ∂G so that ω ≡ 1 or F = ∅ so that ω ≡ 0.
Boundedness of paraproducts on spaces of homogeneous type I
Published in Applicable Analysis, 2022
Der-Chen Chang, Xing Fu, Dachun Yang
It has been proven that many classical results on Euclidean spaces from harmonic analysis still hold true on spaces of homogeneous type, in the sense of Coifman and Weiss [24,25], which are generalizations of many important underlying spaces such as non-isotropic Euclidean spaces, quasi-metric spaces equipped with the Ahlfors n-regular measure, compact Riemannian manifolds equipped with the Euclidean distance and the Lebesgue measure, the boundary of a Lipschitz domain in equipped with the Euclidean distance and the harmonic measure, etc.; see [25, pp. 588-590] for more examples. Recall that a quasi-metric space is a non-empty set equipped with a quasi-metricd such that, for any , ; if and only if x = y;d satisfies the quasi-triangle inequality, where is called the quasi-triangle constant which is independent of x, y, and z.
On increasing stability in an inverse source problem with local boundary data at many wave numbers
Published in Applicable Analysis, 2022
Let be the sector of the complex plane and be the harmonic measure in .