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Function Spaces and Partial Differential EquationsVolume 1 - Classical Analysis$
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Ali Taheri

Print publication date: 2015

Print ISBN-13: 9780198733133

Published to Oxford Scholarship Online: September 2015

DOI: 10.1093/acprof:oso/9780198733133.001.0001

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Convolution Semigroups; The Poisson and Heat Kernels on Rn

Convolution Semigroups; The Poisson and Heat Kernels on Rn

(p.309) 9 Convolution Semigroups; The Poisson and Heat Kernels on Rn
Function Spaces and Partial Differential Equations

Ali Taheri

Oxford University Press

This chapter discusses convolutions on ℝn in detail. The Banach algebra L1 and approximations to identity are introduced and studied. Various applications including Young’s convolution inequality, boundedness of Riesz potentials and Loomis-Whitney inequality are discussed. A detailed discussion of the Cauchy initial value problem for the wave equation on ℝn is also covered at the end.

Keywords:   Poisson kernel, heat kernel, Young’s convolution inequality, Friedrich mollifiers, approximation to identity, Loomis-Whitney inequality, wave equation, Hadamard descent, Duhamel principle

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