2016/05/16 by Paata Ivanisvili, Ivanisvili, Paata · 1 citation
Mathematics · #42B35 #47A30 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:42B35 #msc:47A30
paper · pdf · doi:10.48550/arxiv.1605.04840
25 pages, 2 figures. Fixed several typos. Added a new section (Section 3.1) about how to solve PDE. Removed less valuable corollaries from the paper. Added some references. Slightly modified the abstract
arxiv created 2017/06/20 · arxiv updated 2017/06/22
Let I, J⊂ ℝ be closed intervals, and let H be C3 smooth real valued function on I× J with nonvanishing Hx and Hy. Take any fixed positive numbers a,b, and let dμ be a probability measure with finite moments and absolutely continuous with respect to Lebesgue measure. We show that for the inequality ∫ℝn ess supy ∈ ℝn H( f((x-y)/(a)),g((y)/(b)))dμ(x) ≥ H(∫ℝnfdμ, ∫ℝngdμ) to hold for all Borel functions f,g with values in I and J correspondingly it is necessary that a2\fracHxxHx2+(1-a2-b2)\fracHxyHxHy+b2\fracHyyHy2≥ 0, |a-b|≤ 1, a+b≥ 1 and ∫ℝnxdμ=0 if a+b>1. Moreover, if dμ is a Gaussian measure then the necessary condition becomes sufficient. This extends Prékopa--Leindler and Ehrhard inequalities to an arbitrary function H(x,y). As an immediate application we obtain the new proof of the Ehrhard inequality. In particular, we show that in the class of even probability measures with smooth positive density and finite moments the Gaussian measure is the only one which satisfies the functional form of the Ehrhard inequality on the real line with their own distribution functions.