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Analysis of Heat Equations on Domains. (LMS-31) by El-Maati Ouhabaz

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Chapter Eight

A REVIEW OF THE KATO SQUARE ROOT PROBLEM

T. Kato asked in the early 1960’s the question whether for a divergence form uniformly elliptic operator A; the domain of the square root4111 coincides with the domain of the sesquilinear form of A: The question was solved in few particular cases but the general case was open for several decades, and was known as the Kato square root problem. The problem was solved in the case of dimension one by Coifman, McIntosh, and Meyer [CMM82]. The case of arbitrary dimension has been solved only recently by Hofmann,Lacey, and McIntosh [HLM02] for second-order divergence form operators whose heat kernel ...

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