Kato's square root problem in Banach spaces

2007/03/01 by Tuomas Hytönen, Hytonen, Tuomas, Alan McIntosh +3
Mathematics · #46B09 #46E40 #47A60 #47F05 #60G46 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.math/0703012

openalex publication_date 2007/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L^p(Rⁿ;X) of X-valued functions on Rⁿ. We characterize Kato's square root estimates ‖√(L)u‖_p \eqsim ‖∇ u‖_p and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative L^p space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the L^p theory of square roots of elliptic operators, as well as an L^p version of Carleson's inequality.

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