Probabilistic Sobolev embeddings and applications. (Injections de Sobolev probabilistes et applications.) (French. English summary) Zbl 1296.46031

Summary: In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold \((M,g)\). More precisely, we prove that, for natural probability measures on \(L^2(M)\), almost every function belongs to all spaces \(L^p(M)\), \(p < + \infty\). We then give applications to the study of the growth of the \(L^p\) norms of spherical harmonics on spheres \(\mathbb S^d\): we prove (again for natural probability measures) that almost every Hilbert base of \(L^2(\mathbb S^d)\) made of spherical harmonics has all its elements uniformly bounded in all spaces \(L^p(\mathbb S^d)\), \(p < + \infty\). We also prove similar results on tori \(\mathbb T^d\). We then give an application to the study of the decay rate of damped wave equations in a framework where the geometric control property of C. Bardos et al. [SIAM J. Control Optim. 30, No. 5, 1024–1065 (1992; Zbl 0786.93009)] is not satisfied. Assuming that it is violated for a measure \(0\) set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the \(H^1\)-supercritical wave equation, for which we prove that, for almost all initial data, the weak solutions are strong and unique, locally in time.


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35L70 Second-order nonlinear hyperbolic equations


Zbl 0786.93009
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