2019/08/01 by Evan Randles, Randles, Evan, Laurent Saloff‐Coste +1 · 1 citation
Computer Science · Mathematics · #35H30 (Secondary) #35K08 (Primary) 35K25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1908.00595
openalex publication_date 2019/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a class of constant-coefficient partial differential operators on\na finite-dimensional real vector space which exhibit a natural dilation\ninvariance. Typically, these operators are anisotropic, allowing for different\ndegrees in different directions. The heat kernels associated to these so-called\npositive-homogeneous operators are seen to arise naturally as the limits of\nconvolution powers of complex-valued measures, just as the classical heat\nkernel appears in the central limit theorem. Building on the\nfunctional-analytic approach developed by E. B. Davies for higher-order\nuniformly elliptic operators with measurable coefficients, we formulate a\ngeneral theory for (anisotropic) self-adjoint variable-coefficient operators,\neach comparable to a positive-homogeneous operator, and study their associated\nheat kernels. Specifically, under three abstract hypotheses, we show that the\nheat kernels satisfy off-diagonal (Gaussian type) estimates involving the\nLegendre-Fenchel transform of the operator's principle symbol. Our results\nextend those of E. B. Davies and G. Barbatis and partially extend results of A.\nF. M. ter Elst and D. Robinson.\n