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A Convexity Theorem for Local mean values of Subtemperatures

A Convexity Theorem for Local mean values of Subtemperatures A CONVEXITY THEOREM FOR LOCAL MEAN VALUES OF SUBTEMPERATURES N. A. WATSON 1. Introduction Subharmonic functions are characterized by an inequality involving their integral means over spheres, and there is no question that spheres are the best surfaces to use. Subsolutions of the heat equation, however, have been characterized in an analogous manner using means over two different surfaces. Usually the boundaries of rectangles are used, as in [3, p. 277], and these have the advantage that a Poisson kernel for rectangles is known, at least as the sum of a series. However, in [11], level surfaces of the fundamental solution (called fundamental surfaces below) were used, the motivation coming from a mean value theorem for temperatures due to Pini [9] and Fulks [6]. For most purposes the two approaches are equally good; but in establishing a Wiener criterion for the heat equation, the use of fundamenial surfaces is essential [5]. In this paper we study the properties of the means of subtemperatures over fundamental surfaces. The corresponding statements for rectangular surfaces are all either unknown or demonstrably false. Theorem 1 is a linearity theorem for the means of temperatures, analogous to the classical result that the means of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Convexity Theorem for Local mean values of Subtemperatures

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/22.3.245
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Abstract

A CONVEXITY THEOREM FOR LOCAL MEAN VALUES OF SUBTEMPERATURES N. A. WATSON 1. Introduction Subharmonic functions are characterized by an inequality involving their integral means over spheres, and there is no question that spheres are the best surfaces to use. Subsolutions of the heat equation, however, have been characterized in an analogous manner using means over two different surfaces. Usually the boundaries of rectangles are used, as in [3, p. 277], and these have the advantage that a Poisson kernel for rectangles is known, at least as the sum of a series. However, in [11], level surfaces of the fundamental solution (called fundamental surfaces below) were used, the motivation coming from a mean value theorem for temperatures due to Pini [9] and Fulks [6]. For most purposes the two approaches are equally good; but in establishing a Wiener criterion for the heat equation, the use of fundamenial surfaces is essential [5]. In this paper we study the properties of the means of subtemperatures over fundamental surfaces. The corresponding statements for rectangular surfaces are all either unknown or demonstrably false. Theorem 1 is a linearity theorem for the means of temperatures, analogous to the classical result that the means of

Journal

Bulletin of the London Mathematical Society – Wiley

Published: May 1, 1990

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