When a function belonging to fractional-order Sobolev space is supported in proper subset of the Lipschitz domain on which defined, how its norm as smaller set compared whole domain? On what do comparison constants depend on? Do different norms behave differently? This article addresses these issues. We prove some inequalities and disprove misconceptions by counter-examples.

We define Sobolev norms of arbitrary real order for a Banach representation $(\pi, E)$ Lie group, with regard to single differential operator $D=d\pi(R^2+\Delta)$. Here, $\Delta$ is Laplace element in the universal enveloping algebra, and $R>0$ depends explicitly on growth rate representation. In particular, we obtain spectral gap $D$ space smooth vectors $E$. The main tool novel factorization ...

We prove logarithmic Sobolev inequality for measures q(x) = dist(X) = exp ( −V (x) ) , x ∈ R, under the assumptions that: (i) the conditional distributions Qi(·|xj , j 6= i) = dist(Xi|Xj = xj , j 6= i) satisfy a logarithmic Sobolev inequality with a common constant ρ, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian V are not too large r...

This paper presents a new efficient approach for the solution of the lp-lq minimization problem based on the application of successive orthogonal projections onto generalized Krylov subspaces of increasing dimension. The subspaces are generated according to the iteratively reweighted least-squares strategy for the approximation of lp/lq-norms by weighted l2-norms. Computed image restoration exa...

We prove that the family of functionals (Iδ) defined by Iδ(g) = ∫∫ RN×RN |g(x)−g(y)|>δ δ |x− y|N+p dx dy, ∀ g ∈ L(R ), for p ≥ 1 and δ > 0, Γ-converges in L(R ), as δ goes to 0, when p ≥ 1. Hereafter | | denotes the Euclidean norm of R . We also introduce a characterization for BV functions which has some advantages in comparison with the classic one based on the notion of essential variation o...

We study the large time behavior of solutions to the dissipative Korteweg-de Vries equations ut + uxxx + |D|αu + uux = 0 with 0 < α < 2. We find asymptotic expansions of the solution as t→∞ in various Sobolev norms.

We investigate the problem of equivalence Lq-Sobolev norms in Malliavin spaces for q∈[1,∞), focusing on graph norm k-th derivative operator and full Sobolev involving all derivatives up to order k, where k is any positive integer. The case q=1 infinite-dimensional setting challenging, since at such extreme standard approach Meyer's inequalities fails. In this direction, we are able establish me...

We approximate functions defined on smooth bounded domains by elements of the eigenspaces Laplacian or Stokes operator in such a way that approximations are and converge both Sobolev Lebesgue spaces. prove an abstract result referred to fractional power spaces positive, self-adjoint, compact-inverse operators Hilbert spaces, then obtain our main using explicit form these for Dirichlet operators...

We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms. AMS numbers 2000: Primary: 46E35. Secondary: 35R11, 42B35, 58J35.