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An inequality of rearrangements on the unit circle

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Department of Mathematical Sciences, Aalborg University
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  • Earth Science

Abstract

AALBORG UNIVERSITY ' & $ % An inequality of rearrangements on the unit circle by Cristina Draghici R-2005-31 October 2005 Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G DK - 9220 Aalborg Øst Denmark Phone: +45 96 35 80 80 Telefax: +45 98 15 81 29 URL: http://www.math.aau.dk e ISSN 1399–2503 On-line version ISSN 1601–7811 AN INEQUALITY OF REARRANGEMENTS ON THE UNIT CIRCLE CRISTINA DRAGHICI Abstract. We prove that the integral of the product of two functions over a symmetric set in S1 × S1, defined as E = {(x, y) ∈ S1 × S1 : d(σ1(x), σ2(y)) ≤ α}, where σ1, σ2 are diffeomorphisms of S 1 with certain properties and d is the geodesic distance on S1, increases when we pass to their symmetric decreasing rearrangement. We also give a characterization of these diffeomorphisms σ1, σ2 for which the rearrangement inequality holds. As a consequence, we obtain the result for the integral of the function Ψ(f(x), g(y)) (Ψ a supermodular function) with a kernel given as k[d(σ1(x), σ2(y))], with k decreasing. 1. Introduction On a measure space (X,µ), the Hardy-Littlewood inequality asserts [4]:∫ X f(x)g(x) dµ(x) ≤ ∫ µ(X) 0 f∗(t)g∗(t) dt, where f ∗ and g∗ are the decreasing rearrangements of f and g, respectively. In what follows, X = S1, or X = [−pi, pi], and the above inequality can be written as: (1.1) ∫ pi −pi f(x)g(x) dx ≤ ∫ pi −pi f ](x)g](x) dx, with f ], g] the symmetric decreasing rearrangements of f and g, given by f ](x) = f∗(2|x|) and g](x) = g∗(2|x|). These inequalities can be proved using the layer-cake formula [10]: Every mea- surable function f : X → R+ can be written as an integral of the characteristic function of its level sets: (1.2) f(x) = ∫ ∞ 0 χ{f>t}(x) dt. A more general rearrangement inequality on X = Rn is the Riesz-Sobolev inequal- ity: (1.3) ∫ R2n f(x)g(y)h(x− y) dxdy ≤ ∫ R2n f ](x)g](x)h](x− y) dxdy, where f , g, h are non-negative functions which vanish at infinity in a weak sen

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