Non-archimedean generalized Bessel potentials and their applications

2020/09/11 by Torresblanca-Badillo, Anselmo
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2009.05630

Abstract

This article describes a class of pseudo-differential operators (A^αφ)(x)=F^-1_{ξ→ x}([max\|\boldsymbolψ₁(||ξ||_p)|,|\boldsymbolψ₂(||ξ||_p)|\]^-α\widehatφ(ξ)), φ∈ D(ℚ_pⁿ) and α∈ℂ; here [max\|\boldsymbolψ₁(||ξ||_p)|,|\boldsymbolψ₂(||ξ||_p)|\]^-α is the symbol of the operator A^α. These operators can be seen as a generalization of the Bessel potentials in the p-adic context. We show that the family (K_α)_α>0 of convolution kernels attached to generalized Bessel potentials A^α, α>0, determine a convolution semigroup on ℚ_pⁿ. Imposing certain conditions we have that K_α, α>0, is a probability measure on ℚ_pⁿ. Moreover, we will study certain properties corresponding to the Green function of the operator A^α and we show that heat equations, naturally associated to these operators, describes the cooling (or loss of heat) in a given region over time.

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