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The $L^p$-extension problem

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Emiel Lorist
(Delft University of Technology)

Let $T$ be a bounded operator on $L^p(S)$ and $X $ be a Banach space. We ask the question when the prescription $(T \otimes I_X )(f \otimes x) := T f \otimes x$ extends by linearity to a bounded operator on the Bochner space $L^p(S ; X)$. Operators of this form play, for example, an important role in the analysis of parabolic partial differential equations from an evolution equation perspective. In this talk I will first thoroughly introduce the $L^p$-extension problem and prove a few elementary sufficient conditions on $X$ and $ T $ for which the $L^p$-extension problem has an affirmative answer. Moreover I will give some simple counterexamples. Towards the end of the talk I will state a few more recent results related to the $L^p$-extension problem, which come from my own research.