The solution of the Kato problem in two dimensions

By S. Hofmann, A. Mcintosh. 2002

47 citations recorded by the source corpus.

We solve, in two dimensions, the "square root problem of Kato". That is, for L = -div (A(x)N), where A(x) is a 2 x 2 accretive matrix of bounded measurable complex coefficients, we prove that L1/2: L12(R2) ? L2(R2). [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4c64771e65b7c54a36e2e1c1ba691af08acac3c2

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Analyticity of layer potentials and $L^{2}$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty}$ coefficients (2007, result): “ect proof 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”
Quadratic estimates and functional calculi of perturbed Dirac operators (2004, unknown): “This result was proved in a series of papers by Hofmann– McIntosh [18], Auscher–Hofmann–Lewis–Tchamitchian [3], Hofmann– Lacey–McIntosh [17], and, in full generality, by Auscher–Hofmann–Lacey–”
The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability (2015, methodology): “The first is quite natural in this context: a stopping time argument, in the spirit of the proofs of the Kato square root conjecture [Hofmann and McIntosh 2002; Hofmann et al. 2002; Auscher et al. 2002a] (and of local Tb theorems [Christ 1990; Auscher et al.”
On a quadratic estimate related to the Kato conjecture and boundary value problems (2008, methodology): “The kind of stoppingtime argument providing an inequality like (16) leading to (17) is developed for the first time in this context in [18].”
Extrapolation of Carleson measures and the analyticity of Kato's square-root operators (2001, methodology): “It was then observed by M. Lacey [17] that the use of an appropriate sectorial decomposition of C n allows one to extend the argument in [ 12 ] to higher dimensions, assuming that the above-mentioned "Gaussian" property holds.”
$L^p$-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].”