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.     

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space $L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P _m?1(x), where P _m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

Sobolev spaces and approximation by affine spanning systems

We develop conditions on a Sobolev function \(\psi \in W^{m,p}({\mathbb{R}}^d)\) such that if \(\widehat{\psi}(0) = 1\) and ψ satisfies the Strang–Fix conditions to order m ? 1, then a scale averaged approximation formula holds for all \(f \in W^{m,p}({\mathbb{R}}^d)\) :

$ f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^{J} \sum_{k \in {{\mathbb{Z}}}^d} c_{j,k}\psi(a_j x - k) \quad {\rm in} W^{m, p}({{\mathbb{R}}}^d).$

The dilations { a _j } are lacunary, for example a _j =  2 ^j , and the coefficients c _j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in \({W^{m - 1,p}({\mathbb{R}}^d)}\) the scale averaging is unnecessary and one has the simpler formula \(f(x) = \lim_{j \to \infty} \sum_{k \in {\mathbb{Z}}^d} c_{j,k}\psi(a_j x - k)\) . The Strang–Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or “spanning” criteria for the small scale affine system \(\{\psi(a_j x - k) : j > 0, k \in {\mathbb{Z}}^d \}\) in \(W^{m,p}({\mathbb{R}}^d)\) . We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?

     

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.     

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G _n /K _n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N _n nilpotent, in other words pairs (G _n , K _n ) for which G _n /K _n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L ²(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G _n /K _n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L ²(G/K). This allows an interpretation of L ²(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.     

On the Generalized Stochastic Dirichlet Problem—Part I: The Stochastic Weak Maximum Principle

We treat the stochastic Dirichlet problem \(L\lozenge u = h+\nabla f\) in the framework of white noise analysis combined with Sobolev space methods. The input data and the boundary condition are generalized stochastic processes regarded as linear continuous mappings from the Sobolev space \(W_0^{1,2}\) into the Kondratiev space (S)_???1. The operator L is assumed to be strictly elliptic in divergence form \(L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u\). Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes, denoted by \(\lozenge\), is interpreted as the Wick product. In this paper we prove the weak maximum principle for the operator L, which will imply the uniqueness of the solution to \(L\lozenge u = h+\nabla f\).     

Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(? ₁, ..., ? _N ) generated by finitely many compactly supported functions ? ₁, ..., ? _N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(? ₁, ..., ? _N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(?) generated by a compactly supported refinable function ?, we prove that for almost all \((x_0, x_1)\in [0,1]^2\), any signal in V(?) can be locally reconstructed from its samples from \(\{x_0, x_1\}+{\mathbb Z}\) with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(?) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.     

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.     

Spherical means in annular regions in the <Emphasis Type="Italic">n</Emphasis>-dimensional real hyperbolic spaces

Let Z _r,R be the class of all continuous functions f on the annulus Ann(r, R) in the real hyperbolic space \(\mathbb B^n\) with spherical means M _s f(x)?=?0, whenever s?>?0 and \(x\in\mathbb B^n\) are such that the sphere S _s (x)???Ann(r, R) and \(B_r(o)\subseteq B_s(x).\) In this article, we give a characterization for functions in Z _r,R . In the case R?=?∞, this result gives a new proof of Helgason’s support theorem for spherical means in the real hyperbolic spaces.     

Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of \(C_0(X)\), for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces.     

On the Generalized Stochastic Dirichlet Problem—Part II: Solvability,Stability and the Colombeau Case

In this paper we consider the stochastic Dirichlet problem \(L\lozenge u=h+\nabla f\) in the framework of white noise analysis combined with Sobolev space and Colombeau algebra methods. The operator L is assumed to be strictly elliptic in divergence form \(L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u\). Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes is interpreted as the Wick product. Generalized random processes are considered as linear bounded mappings from the Sobolev space \(W_0^{1,2}\) into the Kondratiev space (S)_???1. In this paper we prove existence and uniqueness of the problem of this form in the case when the operator L generates a coercive bilinear form, and then extend this result to the general case. We also consider the case when the coefficients of L, the input data and the boundary condition are Colombeau-type generalized stochastic processes.     

The extension of functions of Sobolev classes beyond the boundary of the domain on Carnot groups

We prove the theorem on extension of the functions of the Sobolev space W _p ^l (Ω) which are defined on a bounded (ε, δ)-domain Ω in a two-step Carnot group beyond the boundary of the domain of definition. This theorem generalizes the well-known extension theorem by P. Jones for domains of the Euclidean space.     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Existence of indecomposable members in indecomposable spaces of operators

In this paper we prove that if S is a set of operators acting on a separable L _p -space X, 1 ≤ p < ∞ (or, more generally, on any separable Köthe function space) such that S is indecomposable (that is, no non-trivial subspace of X of the form L _p (A), where A is measurable, is a common S-invariant subspace), then \(\overline {span} \) S admits an indecomposable operator. As applications, we obtain some new results about transitive algebas on separable Hilbert spaces, as well as an extension of the simultaneous Wielandt theorem to semigroups of operators acting on separable L _p -spaces.     

Blow-up solutions for Hardy–Sobolev equations on compact Riemannian manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension \(n\ge 6\), \(\xi _0\in M\), and we are concerned with the following Hardy–Sobolev elliptic equations:

$$\begin{aligned} -\Delta _gu+h(x)u=\frac{u^{2^{*}(s)-1-\epsilon }}{d_{g}(x,\xi _0)^s},\ \ \ \ u>0\ \ \mathrm{in} \ \ M, \end{aligned}$$

(0.1)

where \(\Delta _g\,=\,\mathrm{div}_g(\nabla )\) is the Laplace–Beltrami operator on M, h(x) is a \(C^1\) function on M, \(\epsilon \) is a sufficiently small real parameter, \(2^{*}(s):=\frac{2(n-s)}{n-2}\) is the critical Hardy–Sobolev exponent with \(s\in (0,2)\), and \(d_{g}\) is the Riemannian distance on M. Performing the Lyapunov–Schmidt reduction procedure, we obtain the existence of blow-up families of positive solutions of problem (0.1).

     

Generalized fractional integrals in <Emphasis Type="Italic">p</Emphasis>-adic morrey and Herz spaces

For Riesz potential I ^β (f) on p-adic linear space Q _p ⁿ and its modification \(\widetilde{I^\beta }(f)\) we give sufficient conditions of their boundedness from radialMorrey space to anotherMorrey or Campanato space. Also we study the boundedness of modified Riesz potential \(\widetilde{I^\beta }(f)\) from Herz space to special Campanato spaces.     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

Several extremal approximation problems for the characteristic function of a spherical layer

We discuss three interrelated extremal problems on the set P _n,m of algebraic polynomials of a given degree n on the unit sphere \(\mathbb{S}^{m - 1}\) of the Euclidean space ? ^m of dimension m ≥ 2. (1) Find the norm of the functional \(F\left( \eta \right) = F_h P_n = \int_{\mathbb{G}\left( \eta \right)} {P_n (x)dx}\), which is the integral over the spherical layer \(\mathbb{G}\left( \eta \right) = \left\{ {x = \left( {x_1 , \ldots ,x_m } \right) \in \mathbb{S}^{m - 1} :h' \leqslant x_m \leqslant h''} \right\}\) defined by a pair of real numbers η = (h′, h″), ?1 ≤ h′ < h″ ≤ 1, on the set P _n,m with the norm of the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of functions summable on the sphere. (2) Find the best approximation in \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) of the characteristic function χ _η of the layer \(\mathbb{G}\left( \eta \right)\) by the subspace P _n,m ^⊥ of functions from \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) that are orthogonal to the space of polynomials P _n,m . (3) Find the best approximation in the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of the function χ _η by the space of polynomials P _n,m . We present a solution of all three problems for the values h′ and h″ that are neighboring roots of the polynomial in a single variable of degree n + 1 that deviates least from zero in the space L ₁ ^φ (?1, 1) of functions summable on the interval (?1, 1) with ultraspherical weight φ(t) = (1 ? t ²) ^α , α = (m ? 3)/2. We study the respective one-dimensional problems in the space of functions summable on (?1, 1) with an arbitrary not necessarily ultraspherical weight.