Operators on Bochner spaces

Abstract

Here we bring in the colourful cast of operators acting on the spaces that we study. The first two sections have a general character: we discuss the fundamental problem of extending an operator from an L ^p-space to the corresponding Bochner space, and study different interpolation techniques for operators on these spaces. The next three sections are concerned with distinguished particular operators of classical analysis (the Hardy–Littlewood maximal operator, the Fourier transform, and partial derivatives) on Bochner spaces L ^p(ℝ ^d;X) with a Euclidean domain. In the final section, returning to abstract measure spaces, we develop the theory of conditional expectations, a necessary prerequisite for the discussion of martingales in the subsequent chapter.