Embedding of Sobolev spaces with limit exponent revisited

An embedding of the Sobolev spaces W _p ^s (? ⁿ ) in Lizorkin-type spaces of locally integrable functions of smoothness zero is obtained; a similar assertion for Riesz and Bessel potentials is presented. The embedding theorem is extended to Sobolev spaces on irregular domains in n-dimensional Euclidean space. The statement of the theorem depends on geometric parameters of the domain of functions.     

Embedding constants for periodic Sobolev spaces of fractional order

We obtain an explicit expression for the norms of the embedding operators of the periodic Sobolev spaces into the space of continuous functions (the norms of this type are usually called embedding constants). The corresponding formulas for the embedding constants express them in terms of the values of the well-known Epstein zeta function which depends on the smoothness exponent s of the spaces under study and the dimension n of the space of independent variables. We establish that the embeddings under consideration have the embedding functions coinciding up to an additive constant and a scalar factor with the values of the corresponding Epstein zeta function. We find the asymptotics of the embedding constants as s → n/2.     

Some notes on embedding for anisotropic Sobolev spaces

In this paper, we prove new embedding theorems for generalized anisotropic Sobolev spaces, \(W_{{\Lambda ^{p,q}}(w)}^{{r_1}, \cdots ,{r_n}}\) and \(W_X^{{r_1}, \cdots ,{r_n}}\), where Λ ^p,q (w) is the weighted Lorentz space and X is a rearrangement invariant space in ? ⁿ . The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of B _p weights.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

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.     

Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

The present paper deals with the study of semilinear and non-homogeneous Schrödinger equations on a manifold with conical singularity. We provide a suitable constant by Sobolev embedding constant and for p ∈ (2, 2?) with respect to non-homogeneous term g(x) ∈ L ₂ ^n/2 (B), which helps to find multiple solutions of our problem. More precisely, we prove the existence of two solutions to the problem 1.1 with negative and positive energy in cone Sobolev space H _2,0 ^1,n/2 (B). Finally, we consider p = 2 and we prove the existence and uniqueness of Fuchsian-Poisson problem.     

Spaces of functions of positive smoothness on irregular domains

The paper is devoted to constructing and studying spaces of functions of positive smoothness on irregular domains of the n-dimensional Euclidean space. We prove embedding theorems that connect the spaces introduced with the Sobolev and Lebesgue spaces. The formulations of the theorems depend on geometric parameters of the domain of definition of functions.     

Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ? ∈ W _M ¹ , we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space W _F ¹ . We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.     

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.     

Embedding Theorems for General Multianisotropic Spaces

An integral representation and embedding theorems for functions in multianisotropic Sobolev spaces are proved. Unlike in previous works, the general case where the characteristic Newton polyhedron in ?ⁿ has an arbitrary number of vertices is considered.     

Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups

In this paper, we study the boundedness of the fractional integral operator I _α on Carnot group G in the generalized Morrey spaces M _{p, φ} (G). We shall give a characterization for the strong and weak type boundedness of I _α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.     

Boundary values of differentiable functions defined on an arbitrary domain of a Carnot group

We study the boundary values of the functions of the Sobolev function spaces W _∞ ^l and the Nikol’ski? function spaces H _∞ ^l which are defined on an arbitrary domain of a Carnot group. We obtain some reversible characteristics of the traces of the spaces under consideration on the boundary of the domain of definition and sufficient conditions for extension of the functions of these spaces outside the domain of definition. In some cases these sufficient conditions are necessary.     

On canonical embeddings of complex projective spaces in real projective spaces

In this paper, we construct a natural embedding \(\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} \) of the complex projective space ?P ⁿ considered as a 2n-dimensional, real-analytic manifold in the real projective space \(\mathbb{R}P^{n^2 + 2n} \). The image of the embedding σ is called the ?P ⁿ-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ?P-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ?P _? ⁿ . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ?P _? ⁿ is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.     

Relative widths of Sobolev classes in the uniform and integral metrics

Let W_p^r be the Sobolev class consisting of 2π-periodic functions f such that ‖f^(r)‖_p ≤ 1. We consider the relative widths d_n(W_p^r, MW_p^r, L_p), which characterize the best approximation of the class W_p^r in the space L_p by linear subspaces for which (in contrast to Kolmogorov widths) it is additionally required that the approximating functions g should lie in MW_p^r, i.e., ‖g^(r)‖_p ≤ M. We establish estimates for the relative widths in the cases of p = 1 and p = ∞; it follows from these estimates that for almost optimal (with error at most Cn^?r, where C is an absolute constant) approximations of the class W_p^r by linear 2n-dimensional spaces, the norms of the rth derivatives of some approximating functions are not less than cln min(n, r) for large n and r.     

In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r ^α for r∈(0,1] and any point x from metric space.     

On the traces of Sobolev functions on Lipschitz surfaces

Functions from the Sobolev spaces W _p ¹(Q) are considered on a unit cube Q ? R ⁿ , and the properties of their traces on Lipschitz surfaces are examined. The relation is found between the Hölder exponent α and the Hausdorff dimension of the family of poor k-dimensional planes Γ on which the traces do not belong to C ^α(Γ). For the corresponding families of poor k-dimensional Lipschitz surfaces, estimates in terms of p-modules are obtained.     

On the total-variation convergence of regularizing algorithms for ill-posed problems

It is well known that ill-posed problems in the space V[a, b] of functions of bounded variation cannot generally be regularized and the approximate solutions do not converge to the exact one with respect to the variation. However, this convergence can be achieved on separable subspaces of V[a, b]. It is shown that the Sobolev spaces W ₁ ^m [a, b], m ∈ ? can be used as such subspaces. The classes of regularizing functionals are indicated that guarantee that the approximate solutions produced by the Tikhonov variational scheme for ill-posed problems converge with respect to the norm of W ₁ ^m [a, b]. In turn, this ensures the convergence of the approximate solutions with respect to the variation and the higher order total variations.     

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.