L^p-asymptotic behaviour of solutions of the heat equation on Riemannian symmetric spaces of noncompact type

2024/04/15 by Muna Naik, Naik, Muna, Swagato K. Ray +3
Mathematics · #35B40 #58J35 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Primary 43A85 #Secondary 22E30

paper · pdf · doi:10.48550/arxiv.2404.09985

openalex publication_date 2024/04/15 · openalex created_date 2024/04/17 · openalex updated_date 2026/08/01

Abstract

For Riemannian symmetric spaces X=G/K of noncompact type, we show that for all left K-invariant f∈ L¹(X), the functions ‖h_t‖_L^p(X)^-1(f∗ h_t-M_p(f)h_t) (with h_t being the heat kernel of X) converges to zero in L^p(X), p∈ [1,∞], as t→∞, with the constant M_p(f) depending only on p and f. We also prove an analogous result for the fractional heat kernels h_t^α, α∈ (0,1). The above results have recently been proved for the important special cases p=1 and α= 1,\frac 12.

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