The Solution of the Kato Problem for Divergence Form Elliptic Operators with Gaussian Heat Kernel Bounds

By S. Hofmann, Michael T. Lacey, A. Mcintosh. 2002

83 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1404615743b1dc1cf6c9b9c341d334c8414526df

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Citations in the corpus

$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets (2013, background): “More recently, square function estimates have played an important role in the solution of the Kato problem in [46], [42], [4].”
Quadratic estimates and functional calculi of perturbed Dirac operators (2004, unknown): “ed in Section 2. By Proposition 4.8, this then suﬃces to prove Theorems 2.7 and 2.10. This section is an adaptation of the proof of the Kato square root problem for divergence-form elliptic operators [16, 2, 3], though some estimates require new procedures. For example, we develop new methods based ”
Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators (2012, methodology): “ese preliminaries with some estimate for square functions and non-tangential maximal functions built from the “ellipticized” heat semigroup operators Pt and P∗ t . By the solution of the Kato problem [HLMc], [AHLMcT], we have for every α>0 that (2.11) Z Rn " |x−y|<αt |tPt divx f(y)|2 dydt”
Carleson measures, trees, extrapolation, and T(b) theorems (2001, background): “Indeed, one of the present authors (Auscher), along with Tchamitchian [5], formulated a version of the T(b) Theorem whose proof was based upon the argument of [50], and which was subsequently used to solve the Kato problem in higher dimensions [1], [31], [ 2 ].”
The solution of the Kato problem in two dimensions (2002, background): “The latter result will appear in our forthcoming joint papers with M. T. Lacey [ 9 ], and with Auscher, Lacey and Tchamitchian [1].”
The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability (2015, methodology): “C , ∀Q ∈ DF,Q 0 , where C > 1 depends only on η, β, C1, n and ADR. Proof. The proof is based on a stopping time argument similar to those used in the proof of the Kato square root conjecture [HMc],[HLMc], [AHLMcT], and in local Tb theorems. We begin by noting that (4.15) kMµkL1,∞(σ) := sup λ>0”
The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $ (2001, methodology): “The next step is to use a “T(b)” argument as in [2], [12].”
On a quadratic estimate related to the Kato conjecture and boundary value problems (2008, methodology): “The conical decomposition done in the space of constants (C) to estimate the Carleson measure associated to γt(x) was the main new ingredient of [17].”
Analyticity of layer potentials and $L^{2}$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty}$ coefficients (2007, result): “of of the present formulation of Theorem 8.5 may be found in [A2] or [H2], although we note that the theorem and its proof were already implicit in the proof of the Kato square root conjecture [HMc], [HLMc] and [AHLMcT]; see also the works [Ch], [S] and [AT] for some important antecedents. We shall ”
The solution of the Kato square root problem for second order elliptic operators on Rn (2002, methodology): “To obtain (4.6), we adapt the construction of [22] to verify a variant of the T(b) theorem for square roots [10, Chapter 3, Theorem 3].”