Hilbert, Dirichlet and Fejer families of operators arising from C_0-groups, cosine functions and holomorphic semigroups

Abstract

We extend definitions of Hilbert transform, Dirichlet and Fejer operators from convolution with kernels in Lebesgue spaces to Banach spaces. In these terms the geometric property UMD is characterized.

A unified approach, and relationship to group and semigroup cases, are presented.

Keywords

• Hilbert
• Dirichlet
• and
• Fejer
• families
• operators
• arising
• from
• C_0-groups
• cosine
• functions
• and
• hholomorphic
• semigroups