Search results for "Sobolev"
showing 10 items of 199 documents
Mappings of finite distortion: Monotonicity and continuity
2001
We study mappings f = ( f1, ..., fn) : Ω → Rn in the Sobolev space W loc (Ω,R n), where Ω is a connected, open subset of Rn with n ≥ 2. Thus, for almost every x ∈ Ω, we can speak of the linear transformation D f(x) : Rn → Rn, called differential of f at x. Its norm is defined by |D f(x)| = sup{|D f(x)h| : h ∈ Sn−1}. We shall often identify D f(x) with its matrix, and denote by J(x, f ) = det D f(x) the Jacobian determinant. Thus, using the language of differential forms, we can write
On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝn
2010
AbstractWe establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.
Norm continuity and related notions for semigroups on Banach spaces
1996
We find some conditions on a c0-semigroup on a Banach space and its resolvent connected with the norm continuity of the semigroup. We use them to get characterizations of norm continuous, eventually norm continuous and eventually compact semigroups on Hilbert spaces in terms of the growth of the resolvent of their generator.
Estimates of maximal functions measuring local smoothness
1999
Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I n ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t α (0<α≤1) this definition was given by A.Calderon. In the paper we prove estimates of the maximal functions $${\mathcal{N}}_\eta f$$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_…
Locally Supported Wavelets on Manifolds with Applications to the 2D Sphere
1999
Abstract In this paper we present a construction principle for locally supported wavelets on manifolds once a multiresolution analysis is given. The wavelets provide a stable (or unconditional) basis for a scale of Sobolev spaces H s , 0 ≤ s ≤ s . We examine a fast wavelet transform with almost optimal complexity. For the two-dimensional sphere we construct a multiresolution analysis generated by continuous splines that are bilinear with respect to some special spherical grid. In our approach the poles are not exceptional points concerning the approximation power or the stability of the wavelet basis. Finally we present some numerical applications to singularity detection and the analysis o…
Regularity of the inverse of a Sobolev homeomorphism in space
2006
Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.
Removable sets for Sobolev spaces
1999
We study removable sets for the Sobolev space W1,p. We show that removability for sets lying in a hyperplane is essentially determined by their thickness measured in terms of a concept of p-porosity.
Invertibility of Sobolev mappings under minimal hypotheses
2010
Abstract We prove a version of the Inverse Function Theorem for continuous weakly differentiable mappings. Namely, a nonconstant W 1 , n mapping is a local homeomorphism if it has integrable inner distortion function and satisfies a certain differential inclusion. The integrability assumption is shown to be optimal.
Maximal potentials, maximal singular integrals, and the spherical maximal function
2014
We introduce a notion of maximal potentials and we prove that they form bounded operators from L to the homogeneous Sobolev space Ẇ 1,p for all n/(n − 1) < p < n. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.
Regularity of the Inverse of a Sobolev Homeomorphism
2011
We give necessary and sufficient conditions for the inverse ofa Sobolev homeomorphism to be a Sobolev homeomorphism and conditions under which the inverse is of bounded variation.