The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

Closed embeddings of Hilbert spaces

Motivated by questions related to embeddings of homogeneous Sobolev spaces and to comparison of function spaces and operator ranges, we introduce the notion of closely embedded Hilbert spaces as an extension of that of continuous embedding of Hilbert spaces. We show that this notion is a special case of that of Hilbert spaces induced by unbounded positive selfadjoint operators that corresponds to kernel operators in the sense of L. Schwartz. Certain canonical representations and characterizations of uniqueness of closed embeddings are obtained. We exemplify these constructions by closed, but not continuous, embeddings of Hilbert spaces of holomorphic functions. An application to the closed embedding of a homogeneous Sobolev space on Rn in L₂(Rn), based on the singular integral operator associated to the Riesz potential, and a comparison to the case of the singular integral operator associated to the Bessel potential are also presented. As a second application we show that a closed embedding of two operator ranges corresponds to absolute continuity, in the sense of T. Ando, of the corresponding kernel operators.     

On a class of similarity solutions of the porous media equation,III

It is shown how existence questions for general multiparameter eigenvalue problems can be treated quite simply using degree theory. The equations to be solved are W_n(λ)x_n = 0 ≠ x_n, n = 1, 2,…, k, where λ ? $\text{R}$^k and each W_n(λ) is a self-adjoint linear operator on a Hilbert space H_n. The W_n, which may be unbounded, depend continuously on λ in a suitable sense. A coercivity condition for large ∥ λ ∥ is used, and is shown to be equivalent, in the “linear” case, to a standard determinantal definiteness condition.     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

The weighted Sobolev-Lions type spaces W pl,γ(Ω; E0, E) = W pl,γ(Ω; E) ∩ Lp,γ (Ω; E0) are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω∈ Rn in W pl,γ(; E0, E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ(Ω; E0, E) to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.     

Spectral and Scattering Theory for teh Linear Boltzmann Equation in Exterior Domain

It is well-known that functions u ? W^m,p (Ω) can be extended by a bounded linear operator E to functions Eu ≦ W^m,p( R ⁿ), if Ω is C^M-regular and m ≦ M. Here we prove a corresponding result for grid-functions with extension operators E_h converging to E and mention some applications.     

Numerical ranges and dilations ∗

Denote by W(A) the numerical range of a bounded linear operator A. For two operators A and B (which may act on different Hilbert spaces), we study the relation between the inclusion relation W(A)?W(B) and the condition that A can be dilated to an operator of the form B?I. We also investigate the possibilities of dilating an operator A to operators with simple structure under the assumption that W(A) is included in a special region.     

Composition operators induced by smooth self-maps of the real or complex unit balls

In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.     

Convergence parameter associated with a Markov chain and a family of functions

The proposed definition of convergence parameter R(W) corresponding to a Markov chain X with a measurable state space (E,?) and any nonempty setW of bounded below measurable functions f: E → ? is wider than the well-known definition of convergence parameter R in the sense of Tweedie or Nummelin. Very often, R(W) < ∞, and there exists a set playing the role of the absorbing set inNummelin’s definition ofR. Special attention is paid to the case in whichE is locally compact, X is a Feller chain on E, and W coincides with the family ? ₀ ⁺ of all compactly supported continuous functions f ≥ 0 (f ? 0). In particular, certain conditions for R(? ₀ ⁺ )^?1 to coincide with the norm of an appropriate modification of the chain transition operator are found.     

Function spaces between BMO and critical Sobolev spaces

The function spaces Dk(Rn) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces Ws,p(Rn), where sp=n, obtained by J. Bourgain, H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539-543 (see also J. Van Schaftingen, Estimates for L¹-vector fields, C. R. Math. Acad. Sci. Paris 339 (3) (2004) 181-186). The spaces Dk(Rn) contain all the critical Sobolev spaces. They are embedded in BMO(Rn), but not in VMO(Rn). Moreover, they have some extension and trace properties that BMO(Rn) does not have.     

A W-transform-based criterion for the existence of bounded extensions of E-operators

Let Γ(H) be the symmetric Fock space over a Hilbert space H and ε:H→Γ(H) the exponential mapping. By an E-operator we mean an operator defined on ε(H). For an E-operator A, the composition mapping Φ=A°ε is called its W-transform. In this paper, we obtain a criterion based on the W-transform for checking whether or not an E-operator becomes a bounded linear operator on the Fock space.     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.     

Representations for eigenfunctions of expected value operators in the Wishart distribution

Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the ‘EP’ functions f(S) for which Ef(S) for which E(f(S)) = λ_nf(Σ) for some λ_n ? $\text{R}$, whenever the m × m matrix S has the Wishart distribution W(m, n, Σ). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain L^p spaces.     

Solution to König's Graph Embedding Problem

Let H : L_p ( R ) → L_p( R ), 1 < p < ∞ be the real HILBERT transform. A bounded, linear operator u:E → F (E, F BANACH spaces) is a HT-operator, if the mapping u ? H : E ? L₂( R , E) → L₂( R , F) has a bounded, linear extension to L₂( R ) → L₂( R , F). For E = F and u = id_E BOURGAIN [3] and BURKHOLDER [5] have shown that this holds if and only if E ? UMD. We study these HT-operators and, in particular, we construct a HT-operator which is not UMD-factorable. Furthermore, we show that a UMD-space E is a HILBERT space if and only if |id_E ? H| = 1.     

On operator valued sequences of multipliers and R-boundedness

In recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812-827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441-452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166-186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629-644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E(X) and F(Y) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c₀₀(X) and c₀₀(Y) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators (un)⊂L(X,Y) is called a multiplier sequence from E(X) to F(Y) if the linear operator from c₀₀(X) into c₀₀(Y) which maps (xi)∈c₀₀(X) onto (unxn)∈c₀₀(Y) is bounded with respect to the norms on E(X) and F(Y), respectively. Several cases where E(X) and F(Y) are different (classical) spaces of sequences, including, for instance, the spaces Rad(X) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi-R-bounded and weak-R-bounded from recent papers are also considered in this context.     

Bounded point evaluations and capacity

Let E be a compact set in the complex plane with positive Lebesgue measure, and denote by R^p(E), p ? 1, the closure in the L^p(E) norm of the rational functions with poles off E. A point z?E is said to be a bounded point evaluation for R^p(E) if the map z   ?(z), defined for the rational functions, can be extended to a bounded linear functional on R^p(E). For p < 2 there are no other bounded point evaluations for R^p(E) than the interior points of E, but for p ? 2 there may be bounded point evaluations on the boundary, ∂E. We give a condition, in terms of capacity, which is necessary and sufficient for a point on ∂E to be a bounded point evaluation for R^p(E), 2 < p < ∞, and close to necessary and sufficient when p = 2. We also treat bounded point derivations, and the corresponding problems for L_p-spaces of analytic functions on open sets.     

A local smoothing estimate for the Schrödinger equation

We show that the Schrödinger operator eitΔ is bounded from Wα,q(Rn) to Lq(Rn×[0,1]) for all α>2n(1/2−1/q)−2/q and q?2+4/(n+1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0<t<1|eitΔf| is bounded from Hs(Rn) to when s>s₀ if and only if it is bounded from Hs(Rn) to L²(Rn) when s>2s₀. A corollary is that sup0<t<1|eitΔf| is bounded from Hs(R²) to L²(R²) when s>3/4.     

Integral representation of the W-weighted Drazin inverse for Hilbert space operators

This paper studies the integral representation of the W-weighted Drazin inverse for bounded linear operators between Hilbert spaces. By using operator matrix blocks, some integral representations of the W-weighted Drazin inverse for Hilbert space operators are established.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.