To the Theory of Sobolev-Type Classes of Functions with Values in a Metric Space

We study some classes of functions with values in a complete metric space which can be considered as analogs of the Sobolev spaces W _p ¹ . Earlier the author considered the case of functions on a domain of ? ⁿ . Here we study the general case of mappings on an arbitrary Lipschitz manifold. We give necessary auxiliary facts, consider some examples, and describe some methods of construction of lower semicontinuous functionals on the classes W _p ¹ (M), where M is a Lipschitz manifold.     

On the traces of Sobolev functions on the boundary of a cusp with a Hölder singularity

For the Sobolev classes W _p ¹ on a “zero” cusp with a Hölder singularity at the vertex, we consider the question of compactness of the embedding of the traces of Sobolev functions into the Lebesgue classes on the boundary of the cusp.     

Sur une décomposition topologique des fonctions de Sobolev et sur la localisation de l'énergie dans les plasmas idéaux

Motivated by properties of “frozen topological invariants” in ideal plasmas theory, we consider the problem of an intrinsic characterization of the weak closure in H₀¹ (Ω) of all functions ψ₀ ∘ ϕ, with ϕ : Ω → Ω a diffeomorphism. We show that one may associate with ψ₀ a class W^weak (ψ₀) of functions which captures robust topological properties, of the level sets of {ψ₀} and, moreover, is closed under weak limits.     

Quantum integration in Sobolev classes

We study high-dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes W_p^r([0,1]^d) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Novak on integration of functions from Hölder classes.     

Kolmogorov widths of weighted Sobolev classes on closed intervals

In this paper, we estimate the asymptotics of the Kolmogorov widths of weighted Sobolev classes in the metric of L _p . We establish the relationship between the width of the set W _∞,g ¹ and the approximation of the antiderivative function g by piecewise constant functions.     

Quantum complexity of the approximation for the classes B(Wp ([0, 1])) and B(Hp ([0, 1]))

We study the approximation of functions from anisotropic Sobolev classes B(W^r_p([0,1]d)) and Hölder-Nikolskii classes B(H^r_p([0,1]d)) in the L_q([0,1]d) norm with q ≤ p in the quantum model of computation. We determine the quantum query complexity of this problem up to logarithmic factors. It shows that the quantum algorithms are significantly better than the classical deterministic or randomized algorithms.     

Truncation and aliasing errors for Whittaker-Kotelnikov-Shannon sampling expansion

Let B _?? ^p , 1 ?? p < ??, be the space of all bounded functions from L _p (?) which can be extended to entire functions of exponential type ??. The uniform error bounds for truncated Whittaker-Kotelnikov-Shannon series based on local sampling are derived for functions f ?? B _?? ^p without decay assumption at infinity. Then the optimal bounds of the aliasing error and truncation error of Whittaker-Kotelnikov-Shannon expansion for non-bandlimited functions from Sobolev classes U(W _p ^r (?)) are determined up to a logarithmic factor.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and general-ized Sobolev classes into Lq([0,1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0,1]d)) to Lq([0,1]d) space for all 1 q,p ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.     

Traces of functions of generalized Sobolev classes

Considering the Sobolev type function classes on a metric space equipped with a Borel measure we address the question of compactness of embeddings of the space of traces into Lebesgue spaces on the sets of less “dimension.” Also, we obtain compactness conditions for embeddings of the traces of the classical Sobolev spaces W _p ¹ on the “zero” cusp with a Hölder singularity at the vertex.     

To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W _p ^s and the space B _pq ^s (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.     

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), ${\Omega\subset{\mathbb R}^N}The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), W ì \mathbb R^N{\Omega\subset{\mathbb R}^N} , with p = N, that is, the case of Pohozhaev–Trudinger–Moser inequality. Similarly to the case p < N where the loss of compactness in W^1,p(\mathbb R^N){W^{1,p}({\mathbb R}^N)} occurs due to dilation operators u ?t^(N-p)/pu(tx){u {\mapsto}t^{(N-p)/p}u(tx)} , t > 0, and can be accounted for in decompositions of the type of Struwe’s “global compactness” and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W₀^1,N{W_0^{1,N}} over a ball in \mathbb R^N{{\mathbb R}^N} . We give a one-parameter scale of Hardy–Sobolev functionals, a “p = N”-counterpart of the H?lder interpolation scale, for p > N, between the Hardy functional ò\frac|u|^p|x|^p dx{\int \frac{|u|^p}{|x|^p}\,{\rm d}x} and the Sobolev functional ò|u|^pN/(N-mp)  dx{\int |u|^{pN/(N-mp)} \,{\rm d}x} . Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W ^1,N -norms under Hardy–Sobolev constraints.     

Traces of Sobolev functions on the Ahlfors sets of Carnot groups

We prove the converse of the trace theorem for the functions of the Sobolev spaces W _p ^l on a Carnot group on the regular closed subsets called Ahlfors d-sets (the direct trace theorem was obtained in one of our previous publications). The theorem generalizes Johnsson and Wallin’s results for Sobolev functions on the Euclidean space. As a consequence we give a theorem on the boundary values of Sobolev functions on a domain with smooth boundary in a two-step Carnot group. We consider an example of application of the theorems to solvability of the boundary value problem for one partial differential equation.     

On convergence analysis of space homeomorphisms

Various theorems on convergence of general spatial homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring Q-homeomorphisms are obtained. In particular, it is established that a family of all ring Q-homeomorphisms f in ? ⁿ fixing two points is compact provided that the function Q is of finite mean oscillation. The corresponding applications have been given to mappings in the Sobolev classes W _loc ^1,p for the case p > n ? 1.     

Optimal Recovery of Periodic Functions from Fourier Coefficients Given with an Error

We construct optimal methods of recovery of 2π-periodic functions analytic in a strip and its derivatives at a pointt∈ [0, 2π), using information about the Fourier coefficients given with an error in the uniform norm. The same problem is solved for the Sobolev spaceW^r₂.     

The hardy-littlewood maximal function of a sobolev function

We prove that the Hardy-Littlewood maximal operator is bounded in the Sobolev spaceW ^1,p (R ⁿ ) for 1<p≤∞. As an application we study a weak type inequality for the Sobolev capacity. We also prove that the Hardy-Littlewood maximal function of a Sobolev function is quasi-continuous.     

On mappings in the Orlicz–Sobolev classes on Riemannian manifolds

We study the problems of the continuous and homeomorphic extension to the boundary of lower Q-homeomorphisms between domains on Riemannian manifolds and formulate the corresponding consequences for homeomorphisms with finite distortion in the Orlicz–Sobolev classes W_loc^1,j W_{loc}^{1,\varphi } under a condition of the Calderon type for the function φ and, in particular, in the Sobolev classes W_loc^1,p W_{loc}^{1,p} for p > n − 1.     

Kolmogorov widths of Sobolev classes on an irregular domain

We establish asymptotic estimates for the Kolmogorov widths of Sobolev classes W _p ^s (K) in the metric of L _q (K) for a power-law peak K ? ? ^d . These estimates are sharp in the order and coincide with order estimates for the unit cube.