Kato's square root problem in Banach spaces

By Tuomas Hytönen, A. Mcintosh, Pierre Portal. 2007

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Abstract Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L p ( R n ; X ) of X -valued functions on R n . We characterize Kato's square root estimates ‖ L u ‖ p ≂ ‖ ∇ 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/78cc2434a484026679a718c6469b35c828e8d6af

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Citations in the corpus

γ-Radonifying Operators: A Survey (2010, unknown): “In harmonic analysis, γ-radonifying norms are the natural generalisation of the square functions arising in connection with Littlewood-Paley theory (see e.g. Stein [111]) and were used as such in Kalton and Weis [63], Hytönen [49], Hytönen, McIntosh, Portal [50], and Hytönen, van Neerven, Portal ”
The square root problem for second-order, divergence form operators with mixed boundary conditions on Lp (2012, background): “ination of domains for fractional powers in the non-Hilbert space case seem to culminate in [54]. In recent years the problem has been investigated in the case of Lp (p6= 2) for instance in [5], [8], [40], [41], [37], [9]; but only the last three are dedicated to the case of a nonsmooth Ω 6= Rd. In ”
Conical square function estimates in UMD Banach spaces and applications to H∞-functional calculi (2007, background): “However this fruitful linearisation has, so far, been limited to the above “vertical” square functions estimates, leaving aside the “conical” estimates of the form (1.2) ( Rn ( |y−x| t ∣∣t √ ∆e−t √ ∆f(y) ∣∣2 dy dt tn+1 ) p 2 dx ) 1 p h ‖f‖Lp(Rn), 1 p ≤ 2.”