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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.

MSC:

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

Citations:

Zbl 0786.93009
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