Operators with continuous kernels

2019/03/15 by Arendt, W., ter Elst, A. F. M. · 1 citation
#47B10 #47G10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1903.06359

Abstract

Let Ω⊂ \bf R^d be open. We investigate conditions under which an operator T on L₂(Ω) has a continuous kernel K ∈ C( Ω× Ω). In the centre of our interest is the condition T L₂(Ω) ⊂ C( Ω), which one knows for many semigroups generated by elliptic operators. This condition implies that T³ has a kernel in C( Ω× Ω) if T is self-adjoint and Ω is bounded, and the power 3 is best possible. We also analyse Mercer's theorem in our context.

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