Interest in Sobolev type equations has recently increased signi cantly, moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated not so much by the desire to ll up the theory but by the aspiration to comprehend nonclassical models of mathematical physics in quasi-Banach spaces. Notice that the Sobolev type equations are called evolutionary if soluti...

We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the ...

in hilbert space l2(rn), we prove the equivalence between the mod-ulus of smoothness and the k-functionals constructed by the sobolev space cor-responding to the fourier transform. for this purpose, using a spherical meanoperator.

For a wide class of Sobolev type norms with respect to measures with unbounded support on the real line, the contracted zero distribution and the logarithmic asymptotic of the corresponding re-scaled Sobolev orthogonal polynomials is given.

We prove a strong form of the quantitative Sobolev inequality in R for p ≥ 2, where the deficit of a function u ∈ Ẇ 1,p controls ‖∇u−∇v‖Lp for an extremal function v in the Sobolev inequality.

Here is established the global existence of smooth solutions to a generalized nonlinear equation of Schrödinger type in the usual Sobolev spaces H and certain weighted Sobolev spaces by using Leray-Schauder fixed point theorem and delicate a priori estimates.

Hardy–Sobolev–type inequalities associated with the operator L := x · ∇ are established, using an improvement to the Sobolev embedding theorem obtained by M. Ledoux. The analysis involves the determination of the operator semigroup {e−tL∗L}t>0.

In this paper, direct estimate of Sobolev exponent of reenable distributions and its application to the asymptotic estimate of Sobolev exponent of M band Daubechies' scaling functions are considered.

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in t...

Various issues with regard to chaos and recurrence in infinite dimensions are discussed. The doctrine we are trying to derive is that Sobolev spaces over bounded spatial domains do host chaos and recurrence, while Sobolev spaces over unbounded spatial domains are lack of chaos and recurrence. Local Sobolev spaces over unbounded spatial domains can host chaos and are natural phase spaces e.g. fo...