Sobolev type spaces in quantum calculus

Quantitative functional calculus in Sobolev spaces

In the framework of Sobolev (Bessel potential) spaces Hn(R,R or C), we consider the nonlinear Nemytskij operator sending a function x ∈ R 7→ f(x) into a composite function x ∈ R 7→ G(f(x), x). Assuming sufficient smoothness for G, we give a ”tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f , with a coefficient depending on G and on the Ha n...

Poincaré–type Inequalities for Broken Sobolev Spaces

We present two versions of general Poincaré–type inequalities for functions in broken Sobolev spaces, providing bounds for the Lq–norm of a function in terms of its broken H1–norm.

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...

A Haar-rado Type Theorem for Minimizers in Sobolev Spaces

Let ω : [0,+∞[→ [0,+∞[ be a modulus of continuity. Consider the integral functional