Lifting in Sobolev spaces

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...

Lifting of S-valued Maps in Sums of Sobolev Spaces

We describe, in terms of lifting, the closure of smooth S-valued maps in the space W ((−1, 1) ;S). (Here, 0 < s <∞ and 1 ≤ p <∞.) This description follows from an estimate for the phase of smooth maps: let 0 < s < 1, let φ ∈ C∞([−1, 1] ;R) and set u = e. Then we may split φ = φ1 + φ2, where the smooth maps φ1 and φ2 satisfy (∗) |φ1|W s,p ≤ C|u|W s,p and ‖∇φ2‖ Lsp ≤ C|u| p W s,p . (∗) was proved...

Sobolev Spaces

Composition in Fractional Sobolev Spaces

1. Introduction. A classical result about composition in Sobolev spaces asserts that if u ∈ W k,p (Ω)∩L ∞ (Ω) and Φ ∈ C k (R), then Φ • u ∈ W k,p (Ω). Here Ω denotes a smooth bounded domain in R N , k ≥ 1 is an integer and 1 ≤ p < ∞. This result was first proved in [13] with the help of the Gagliardo-Nirenberg inequality [14]. In particular if u ∈ W k,p (Ω) with kp > N and Φ ∈ C k (R) then Φ • ...

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...