Sobolev spaces on warped products

Abstract

We study the structure of Sobolev spaces on the cartesian/warped products of a given metric measure space and an interval. Our main results are: - the characterization of the Sobolev spaces in such products - the proof that, under natural assumptions, the warped products possess the Sobolev-to-Lipschitz property, which is key for geometric applications. The results of this paper have been needed in the recent proof of the `volume-cone-to-metric-cone' property of $\mathsf{RCD}$ spaces obtained by the first author and De Philippis.

Corrected few typos in the previous version and updated the presentation

References in corpus (1)

• Tensorization of Cheeger energies, the space $H^{1,1}$ and the area formula for graphs

Cited by in corpus (9)

• The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spaces
• Rigidity for the spectral gap on $RCD(K,\infty )$-spaces
• Recognizing the flat torus among $RC{D}^{\ast }(0,N)$ spaces via the study of the first cohomology group
• From volume cone to metric cone in the nonsmooth setting
• Regularity of Lagrangian flows over $RC{D}^{\ast }(K,N)$ spaces
• Ricci tensor on smooth metric measure space with boundary
• On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth
• Noncompact RCD(0,N) spaces with linear volume growth
• An almost rigidity Theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth