Heisenberg Hausdorff Dimension of Besicovitch Sets

Laura Venieri

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The Leray measure of nodal sets for random eigenfunctions on the torus

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We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in $d \geq 2$ dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity $𝒩 \to \infty$ . The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction...

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The solution of Kato's conjecture (after Auscher, Hofmann, Lacey, McIntosh and Tchamitchian)

Philippe Tchamitchian (2001)

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Kato’s conjecture, stating that the domain of the square root of any accretive operator $L = - div (A \nabla)$ with bounded measurable coefficients in $ℝ^{n}$ is the Sobolev space $H^{1} (ℝ^{n})$ , i.e. the domain of the underlying sesquilinear form, has recently been obtained by Auscher, Hofmann, Lacey, McIntosh and the author. These notes present the result and explain the strategy of proof.