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The Poisson Integral of a Singular Measure

Published online by Cambridge University Press:  20 November 2018

Patrick Ahern
Patrick Ahern*
Affiliation:
University of Wisconsin — Madison, Madison, Wisconsin
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Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y > 0, its Poisson integral is defined by the formula

where CN is chosen so that

Since σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,

is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 4 , 01 August 1983 , pp. 735 - 749
Copyright
Copyright © Canadian Mathematical Society 1983

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