Closure properties of solutions to heat inequalities

2008/06/12 by Jonathan Bennett, Neal Bez, Bennett, Jonathan +1
Mathematics · #35K99 #44A35 #52A40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.0806.2086

openalex publication_date 2008/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if u₁,u₂ : (0,∞) × \R^d → (0,∞) are sufficiently well-behaved solutions to certain heat inequalities on \R^d then the function u: (0,∞) × \R^d → (0,∞) given by u^1/p=u₁^1/p₁ * u₂^1/p₂ also satisfies a heat inequality of a similar type provided \tfrac1p₁ + \tfrac1p₂ = 1 + \tfrac1p. On iterating, this result leads to an analogous statement concerning n-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp n-fold Young convolution inequality and its reverse form.

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