Spectral dimensions of Krein-Feller operators in higher dimensions

Abstract

We study the Dirichlet and Neumann spectral dimensions of Kreĭn-Feller operators associated with finite non-zero Borel measures $\nu$ supported in the open $d$-dimensional unit cube, $d\ge 2$. The central new ingredient is a monotone dyadic spectral set function ${\mathfrak{J}}_{\nu ,a,b}$, whose value on a cube is defined by taking a supremum over all of its dyadic subcubes. This supremum captures the worst local measure-Sobolev embedding scale. The resulting spectral partition function provides a higher-dimensional extension of the $L^q$-spectrum, which governs the one-dimensional theory developed in our earlier work. If the lower $L^{\infty }$-dimension of $\nu$ is larger than $d-2$, the relevant measure-Sobolev embeddings are compact and the upper Dirichlet and Neumann spectral dimensions coincide with the zero of the spectral partition function. Under an explicit regularity condition on the partition function, both spectral dimensions exist and agree. Below the threshold the form construction fails; at the critical threshold,examples exhibit failure of continuity, continuity without compactness, and compactness with an existing spectral dimension. Applications include absolutely continuous measures, Ahlfors-David regular measures, and self-conformal measures without a separation assumption. In the self-conformal setting the spectral partition function, and hence the spectral dimension, is determined by the $L^q$-spectrum. We also derive two-sided eigenvalue-counting bounds for Ahlfors-David regular measures and construct an example for which the spectral dimension does not exist.

27 pages, 1 figure

Cited by in corpus (1)

• Spectral dimensions of Krein--Feller operators and $L^q$-spectra