Simultaneous Packing and Covering in Sequence Spaces

We adapt a construction of Klee (1981) to find a packing of unit balls in ℓ _p (1≤p<∞) which is efficient in the sense that enlarging the radius of each ball to any R>2^1−1/p covers the whole space. We show that the value 2^1−1/p is optimal. Dedicated to the memory of Victor Klee.     

Regularity of Poisson equation in some logarithmic space

In this note, the regularity of Poisson equation -△u = f with f lying in logarithmic function space Lp(LogL)a(Ω)(1＜p ＜∞, a ∈ R) is studied. The result of the note generalizes the W2,p estimate of Poisson equation in Lp(Ω).     

Representation ofL p-norms and isometric embedding inL p-spaces

For fixed 1≦p<∞ theL _p-semi-norms onR ⁿ are identified with positive linear functionals on the closed linear subspace ofC(R ⁿ ) spanned by the functions |<ξ, ·>| ^p , ξ∈R ⁿ . For every positive linear functional σ, on that space, the function Φ_σ:R ⁿ →R given by Φ_σ is anL _p-semi-norm and the mapping σ→Φ_σ is 1-1 and onto. The closed linear span of |<ξ, ·>| ^p , ξ∈R ⁿ is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in anyL _p unlessp=2. Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley.     

Integral Representations and Approximations for Multivariate Gamma Distributions

Let R be a p×p-correlation matrix with an “m-factorial” inverse R ⁻¹ = D − BB′ with diagonal D minimizing the rank m of B. A new -variate integral representation is given for p-variate gamma distributions belonging to R, which is based on the above decomposition of R ⁻¹ without the restriction D > 0 required in former formulas. This extends the applicability of formulas with small m. For example, every p-variate gamma cdf can be computed by an at most -variate integral if p = 3 or p = 4. Since computation is only feasible for small m, a given R is approximated by an m-factorial R ₀. The cdf belonging to R is approximated by the cdf associated with R ₀ and some additional correction terms with the deviations between R and R ₀.     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

The Cauchy Integral Operator on Weighted Hardy Space

The authors show that the Cauchy integral operator is bounded from Hωp(R1) to hωp(R1) (the weighted local Hardy space). To prove the results, a kind of generalized atoms is introduced and a variant of weighted "Tb theorem" is considered.     

On the characterization of p ‐adic Colombeau–Egorov generalized functions by their point values

We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, point values do not determine elements of the so‐called p ‐adic Colombeau–Egorov algebra ??(?ⁿ _p ) uniquely. We further show in a more general way that for an Egorov algebra ??(M, R) of generalized functions on a locally compact ultrametric space (M, d) taking values in a nontrivial ring, a point value characterization holds if and only if (M, d) is discrete. Finally, following an idea due to M. Kunzinger and M. Oberguggenberger, a generalized point value characterization of ??(M, R) is given. Elements of ??(?ⁿ _p ) are constructed which differ from the p ‐adic δ ‐distribution considered as an element of ??(?ⁿ _p ), yet coincide on point values with the latter. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceR ^n–1×R ₊, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL ^p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.     

Some generalizations on the boundedness of bilinear operators

For 0<p<∞, let H^p(R ⁿ) denote the Lebesgue space for p>1 and the Hardy space for p ≤1. In this paper, the authors study H^p(R ⁿ)×H^q(R ⁿ)→H^r(R ⁿ) mapping properties of bilinear operators given by finite sums of the products of the standard fractional integrals or the standard fractional integral with the Calderón-Zygmund operator. The authors prove that such mapping properties hold if and only if these operators satisfy certain cancellation conditions. Supported by the NNSF and the National Education Comittee of China.     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

The space Ω m p (R d ) and some properties

Let m be a v-moderate function defined on R ^d and let g ∈ L ²(R ^d ). In this work, we defineΩ _m ^p (R ^d ) to be the vector space of f ∈ L _m ² (R ^d ) such that the Gabor transform V ^f belongs to L ^p (R ^2d ), where 1 ≤ p < ∞. We equip it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of Ω _m ^p (R ^d ). We also discuss inclusion properties and obtain the dual space of Ω _m ^p (R ^d ). At the end of this work, we study multipliers from L _w ¹ (R ^d ) into Ω _w ^p (R ^d ) and from Ω _w ^p (R ^d ) into L _w−1 ^∞ (R ^d ), where w is the Beurling weight function. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 139–145, January, 2006.     

A Wiener-Tauberian theorem for Fourier-Jacobi transform

Let f∈L^p(R ⁺, Δ(t)dt), we consider conditions under which the space spanned by generalized translates of f is dense in L^p(R ⁺, Δ(t)dt) in terms of Fourier-Jacobi trans form of f. This generalizes and improves the earlier results of A. Sitaram on semi-simple Lie groups of rank one.     

Average Width and Optimal Interpolation of the Sobolev-Wiener Class Wpq(R) in the Metric Lq(R)

In this paper, we determine the exact value of average n − K width _n(W^r_pq(R), L_q(R)) of Sobolev-Wiener class W^r_pq(R) in the metric L_q(R) for 1 > q ≥ p > ∞ and get the value of _n(W^r_p(R), L_qp(R)) for the dual case. We also solve the optimal interpolation problems of W^r_pq(R) in the metric L_q(R) and W^r_p(R) in the metric L_qp(R) for 1 < q ≤ p < ∞.     

Best Simultaneous Approximation in L(I,E)

Let G be a reflexive subspace of the Banach space E and let L^p(I,E) denote the space of all p-Bochner integrable functions on the interval I=[0,1] with values in E, 1p∞. Given any norm N( , ) on R², N nondecreasing in each coordinate on the set R²₊, we prove that L^p(I,G) is N-simultaneously proximinal in L^p(I,E). Other results are also obtained.     

Kolmogorov diameters in solution spaces of systems of partial differential equations

Let E_p(0) denote the solutions (on 0<R ^N) of a system P(D) of partial differential equations with constant coefficients in a localizable analytically uniform space E (defined on 0). The relative Kolmogorov diameters of the neighbourhoods of 0 in E_p(0) are estimated from above and below, using the fundamental principle of Ehrenpreis. The diametral dimension, of E_p(0) is calculated and it is proved, that E_p(R ^N) and E_p(0) are nonisomorphic for (partially) bounded 0, in special cases.Dedicated to Heinz-Günter Tillmann on the occasion of his 60th birthday     

Universal series induced by approximate identities and some relevant applications

We prove the existence of series ∑a_nψ_n, whose coefficients (a_n) are in ∩_p>1?^p and whose terms (ψ_n) are translates by rational vectors in R^d of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener’s algebra W(C₀,?¹), C_b(R^d), C₀(R^d), L^p(R^d), for every p∈[1,∞), and the space of measurable functions. Applying this theory to particular situations, we establish approximations by such series to solutions of the heat and Laplace equations as well as to probability density functions.     

Calderón-Zygmund Operators on the Predual of a Morrey Space

We consider the boundedness of Calderón-Zygmund operators from H ^{p ,ϕ} (R ⁿ )(the predual of a Morrey space) to h ^{p ,ϕ} (R ⁿ )(the local version of H ^{p ,ϕ} ). We show that Calderón's commutator is bounded from H ^{p ,ϕ} to h ^{p ,ϕ} .     

Groups of Isometries of Hardy-Type Spaces of Analytic Functionals in the Unit Ball of the Space~L p

The tensor structure of spaces L _p (R ⁿ ) of summable functions of several variables is described. A scale of Hardy-type spaces of analytic functionals defined in the unit ball of the space L _p (R ¹) of summable functions of one variable is introduced. One-parameter groups of isometries of such spaces of analytic functionals are investigated.     

Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem

We consider a boundary value problem where f(x) ∈ L_p(R), p ∈ [1,∞] (L_∞(R) ≔ C(R) and 0 ≤ q(x) ∈ L^loc₁( R). Boundary value problem (0.1) is called correctly solvable in the given space L_p(R) if for any f(x) ∈ L_p(R) there is a unique solution y(x) ∞ L_p(R) and the following inequality holds with absolute constant c(p) ∈ (0,∞). We find criteria for correct solvability of the problem (0.1) in L_p(R).