Fourier multipliers and Sobolev spaces on homogeneous spaces related to Gelfand pairs

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Résumé

Let G be a locally compact Hausdorff group and let K be a compact subgroup of G such that (G,K) is a Gelfand pair. This paper is devoted to the study of Fourier multipliers and Sobolev spaces on the homogeneous space G/K. The Fourier multipliers and the Sobolev spaces on G/K are built from a vector-valued Fourier transformation related to unitary representations associated with positive definite spherical functions for the Gelfand pair (G,K). Regarding the Fourier multipliers, we investigate their boundedness while for the Sobolev spaces, we look over continuous embedding results. As an application, we solve the bosonic string equation.

Mots-clés

homogeneous space; Gelfand pair; Fourier multiplier; Sobolev space; bosonic string equation