Sobolev Spaces over Loop Groups

Sobolev Spaces

Spinor Fields over Stochastic Loop Spaces

We give the construction of a line bundle over the based Brownian bridge, as well as the construction of spinor fields over the based and the free Brownian bridge. Introduction The Dirac operator over the free loop space is a very important object for the algebraic topology [24]; its index gives the Witten genus and it can predict the rigidity theorem of Witten: the index of some classical oper...

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...

Lifting in Sobolev Spaces

for some function φ : Ω → R. The objective is to find a lifting φ “as regular as u permits.” For example, if u is continuous one may choose φ to be continuous and if u ∈ C one may also choose φ to be C. A more delicate result asserts that if u ∈ VMO (= vanishing means oscillation), then one may choose φ to be also VMO (see R. Coifman and Y. Meyer [1] and H. Brezis and L. Nirenberg [1]). In this...

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...