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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.