Translation-Invariant Operators on Spaces of Vector-Valued Functions

摘要

The treatise deals with translation-invariant operators on various function spaces (including Besov, Lebesgue-Bochner, and Hardy), where the range space of the functions is a possibly infinite-dimensional Banach space X. The operators are treated both in the convolution form Tf = k asterisk f and in the multiplier form in the frequency representation, Tf = m f, where the kernel k and the multiplier m are allowed to take values in L(X) (bounded linear operators on X). Several applications, most notably the theory of evolution equations, give rise to non-trivial instances of such operators. Verifying the boundedness of operators of this kind has been a long-standing problem whose intimate connection with certain randomized inequalities (the notion of R-boundedness which generalizes classical square-function estimates) has been discovered only recently. The related techniques, which are exploited and developed further in the present work, have proved to be very useful in generalizing various theorems, so far only known in a Hilbert space setting, to the more general framework of UMD Banach spaces.