Topology and Sobolev Spaces

Topology and Sobolev Spaces

with 1 ≤ p <∞. W (M,N) is equipped with the standard metric d(u, v) = ‖u− v‖W1,p . Our main concern is to determine whether or not W (M,N) is path-connected and if not what can be said about its path-connected components, i.e. its W -homotopy classes. We say that u and v are W -homotopic if there is a path u ∈ C([0, 1],W (M,N)) such that u = u and u = v. We denote by ∼p the corresponding equiva...

On some questions of topology for -valued fractional Sobolev spaces

"&('*),+/.0 1 )213&4 a.e. 576 Our main results assert that is path-connected if 8!" :9 while it has the same homotopy classes as ;=<> *? @ if 8!BA 9 . We also present some results and open problems about density of smooth maps in C D E? F . Sobre algunas cuestiones topológicas para espacios de Sobolev fracionarios con valores en GIH Resumen. El propósito de este artı́culo es describir las clase...

Sobolev Spaces

Commuatators and Sobolev Spaces

In this paper, we consider the commutators of fractional integrals ∫ Rd b(x)− b(y) |x− y| f(y)dy on the Sobolev spaces . H s p where b is a locally integrable function and r ∈ (0, d). We establish the equivalence between the boundedness of the commutators and the paraproducts of J.M. Bony. Then, we establish necessary and sufficient conditions for the boundedness of the paraproduit operator fro...

Global automorphic Sobolev spaces

The goal is legitimization of term-wise differentiation of L spectral expansions, so that computations producing a classical outcome are correct. We are fond of L expansions because they are what Plancherel gives. Typically, L expansions are not continuous, much less differentiable, so the issue cannot be proving classical differentiability, which does not hold. To say that L spectral expansion...