Quadratic forms and Sobolev spaces of fractional order

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 • ...

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

Discrete Fractional Radon Transforms and Quadratic Forms

We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove that such discrete operators extend to bounded operators from `p to `q for a certain family of kernels. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of H...

Composition operators acting on Sobolev spaces of fractional order – a survey on sufficient and necessary conditions

What follows is a survey of recent results on sufficient and necessary conditions on composition operators to map one Sobolev space of fractional order into another. This report may be taken as a continuation of the contribution of G.Bourdaud given at the forerunner conference of this one, held in Friedrichroda 1992, cf. [Bo 5]. Composition operators are simple examples of nonlinear operators. ...

Composition Operators Acting on Sobolev Spaces of Fractional Order { a Survey on Suucient and Necessary Conditions