On the continuity of heat potentials

Dont, Miroslav

doi:10.21136/cpm.1981.118085

Cited by 3 publications

(4 citation statements)

References 2 publications

(4 reference statements)

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“…Since both Gν and G(ν − ν) are lower semicontinuous, we see that also the restriction of Gν to K is continuous. By [9,Theorem 5], this implies that Gν is continuous on Ê 2 . So K is not semipolar, a contradiction.…”

Section: If Mmentioning

confidence: 98%

Semipolar Sets and Intrinsic Hausdorff Measure

Hansen

Netuka

2018

Potential Anal

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Given a "Green function" G on a locally compact space X with countable base, a Borel set A in X is called G-semipolar, if there is no measure ν = 0 supported by A such that Gν := G(·, y) dν(y) is a continuous real function on X. Introducing an intrinsic Hausdorff measure m G using G-ballsThis is of interest, since G-semipolar sets are semipolar in the potentialtheoretic sense (countable unions of totally thin sets, hit by a corresponding process at most countably many times) provided G is really a Green function for a harmonic space or, more generally, a balayage space.For classical potential theory and Riesz potentials on Ê n or, more generally, for Green functions on a metric measure space (X, d, µ) (where balls are relatively compact) given by a continuous heat kernel (x, y, t) → p t (x, y) with upper and lower bounds of the form t −α/β Φ j (d(x, y)t −1/β ), j = 1, 2, the intrinsic Hausdorff measure is equivalent to an ordinary Hausdorff measure m α−β .It is shown that for the corresponding space-time situation on X × Ê (heat equation on Ê n × Ê in the classical case of the Gauss-Weierstrass kernel) the intrinsic Hausdorff measure is equivalent to an anisotropic Hausdorff measure m α,β (with α = n and β = 2 for the heat equation).In particular, our result solves an open problem for the heat equation (which was the initial motivation for the paper).

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Section: If Mmentioning

confidence: 98%

Semipolar Sets and Intrinsic Hausdorff Measure

Hansen

Netuka

2018

Potential Anal

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“…Remark on the continuity of thermal potentials. In his paper [3], Dont considers the continuity properties of potentials of measures supported by certain J^-sets, in the case n = 1. In the example which begins on p. 163 of [3] Since a < \, this implies that K^ is an J^-set.…”

Section: W(x O -Yt 0 -S)d/i(ys) ^ R N Cmentioning

confidence: 99%

“…In his paper [3], Dont considers the continuity properties of potentials of measures supported by certain J^-sets, in the case n = 1. In the example which begins on p. 163 of [3] Since a < \, this implies that K^ is an J^-set. Furthermore, the projection of Kj onto the x-axis is ^(D^), which consists of all points in [0,1] which have no binary expansion that contains a run of more than j consecutive zeros.…”

Section: W(x O -Yt 0 -S)d/i(ys) ^ R N Cmentioning

confidence: 99%

A Hausdorff measure classification of polar sets for the heat equation

Taylor

Watson

1985

Math. Proc. Camb. Phil. Soc.

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“…considered on K, where A 0 is a measure on <a, b> derived from the function cp (dA o (0 = d(p(t); note that any measure on <a, b> can be considered a measure in R 2 with support contained in the set K -see [7], for instance).…”

Section: Oxmentioning

confidence: 99%