Equivalent Norms for Sobolev Spaces

A Second Look on Definition and Equivalent Norms of Sobolev Spaces

Sobolev’s original definition of his spaces L(Ω) is revisited. It only assumed that Ω ⊆ n is a domain. With elementary methods, essentially based on Poincaré’s inequality for balls (or cubes), the existence of intermediate derivates of functions u ∈ L(Ω) with respect to appropriate norms, and equivalence of these norms is proved.

Sobolev Spaces

Sobolev Norms of Automorphic Functionals

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the represent...

Affine Synthesis and Coefficient Norms for Lebesgue, Hardy and Sobolev Spaces

The affine synthesis operator Sc = P j>0 P k∈Zd cj,kψj,k is shown to map the mixed-norm sequence space `(`) surjectively onto L(R), 1 ≤ p < ∞, under mild conditions on the synthesizer ψ ∈ L(R) (say, having a radially decreasing L majorant near infinity) and assuming R Rd ψ dx = 1. Here ψj,k(x) = | det aj |ψ(ajx− k), for some dilation matrices aj that expand. Therefore the standard norm on f ∈ L...

Wavelet characterization of Sobolev norms∗

Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. We begin with the classical definition of Sobolev spaces. Definition 1. Let k be a nonnegative integer and let 1 < p < ∞ ...