About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.     

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

Integral Operators Induced by the Fock Kernel

We study the L ^p boundedness and find the norm of a class of integral operators induced by the reproducing kernel of Fock spaces over . Dostanić is partialy supported by MNZZS Grant N°ON144010, Zhu is partially supported by the US National Science Foundation.     

Carleson Measures for Spaces of Dirichlet Type

If 0 < p < ∞ and α > − 1, the space consists of those functions f which are analytic in the unit disc and have the property that f ′ belongs to the weighted Bergman space A_α^p. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in such that the measure μ_g,p on defined by dμ_g,p (z) = (1 − |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces      

Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics for standard pseudo-hermitian spaces H_p,q. Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of H_p,q. We study the topology of the projective quadrics and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2ⁿ⁻¹, 2ⁿ⁻¹). The spinorial group Spin U(p,q) is imbedded into SU(2ⁿ⁻¹, 2ⁿ⁻¹). At last, we study the natural imbeddings of the projective quadrics      

On Negative Inertia of Pick Matrices Associated with Generalized Schur Functions

It is known [6] that for every function f in the generalized Schur class and every nonempty open subset Ω of the unit disk , there exist points z₁,...,z_n ∈Ω such that the n × nPick matrix has κ negative eigenvalues. In this paper we discuss existence of an integer n₀ such that any Pick matrix based on z₁,...,z_n ∈Ω with n ≥ n₀ has κ negative eigenvalues. Definitely, the answer depends on Ω. We prove that if , then such a number n₀ does not exist unless f is a ratio of two finite Blaschke products; in the latter case the minimal value of n₀ can be found. We show also that if the closure of Ω is contained in then such a number n₀ exists for every function f in .     

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

H ∞-calculus for the Stokes operator on unbounded domains

We consider the Stokes operator A on unbounded domains of uniform C ^1,1-type. Recently, it has been shown by Farwig, Kozono and Sohr that – A generates an analytic semigroup in the spaces , 1 < q < ∞, where for q ≥ 2 and for q ∈ (1, 2). Moreover, it was shown that A has maximal L ^p -regularity in these spaces for p ∈ (1,∞). In this paper we show that ɛ + A has a bounded H ^∞-calculus in for all q ∈ (1, ∞) and ɛ > 0. This allows to identify domains of fractional powers of the Stokes operator. Received: 12 October 2007     

Spaces of Operators,the ψ-Daugavet Property,and Numerical Indices

Suppose ψ : [0, ∞) → [1, ∞) is a strictly increasing function. A Banach space X is said to have the ψ-Daugavet Property if the inequality holds for every compact operator T : X → X. We show that, if 1 < p < ∞ and K(ℓ_p)↪ X ↪ B(ℓ_p), then X has the ψ-Daugavet Property with (here and c_p is an absolute constant). We also prove that a C^*-algebra A is commutative if and only if for any . Together, these results allow us to distinguish between some types of von Neumann algebras by considering spaces of operators on them. The author was supported in part by the NSF grant DMS-9970369.     

On the numerical range of a Volterra operator in Lp

LetA be the linear operator inL _p (0, 1), 1<p<∞,p≠2, defined by ,x∈L _p (0, 1),s∈[0,1]. We show that the real values of numbers in the numerical range ofA have maximum , whereq=p/(p−1). This amounts to an inequality between integrals, for which we determine the case of equality.     

Bloch-Bergman Pullbacks with Logarithmic Weights

We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight. For 0 < p < ∞, 1 < α < ∞, let A^{p, log α} denote the space of holomorphic functions F in the unit disc D for which

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

Extremal Problems for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils

Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition where stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V-spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as L_p(0,1) and C_p, are non-V-spaces if p 2; and that the spaces l_p are V-spaces if and only if 1 < p < . The authors pose and discuss some related open problems.     

On a Class of Integral Operators

In this paper we consider the space where dv _s is the Gaussian probability measure. We give necessary and sufficient conditions for the boundedness of some classes of integral operators on these spaces. These operators are generalizations of the classical Bergman projection operator induced by kernel function of Fock spaces over .      

An Analogue of Beurling–Hörmander’s Theorem Associated with Partial Differential Operators

In this paper for a positive real number α we consider two partial differential operators D and D_α on the half–plane We define a generalized Fourier transform associated with the operators D and D_α. We establish an analogue of Beurling–H?rmander’s Theorem for this transform and we give some applications of this theorem.     

The diffusive limit of Carleman-type models in the range of very fast diffusion equations

We improve the existing results on the limiting behavior of the Cauchy problem for a class of Carleman-like models with power-type interaction rate in the diffusive scaling with data in the spaces L ^p , 1 ≤ p ≤ ∞. The convergence result, which has been carefully established before for exponents of the interaction rate α ≤ 1, is extended here to the range of exponents 1 < α < 4/3. In addition, we discuss the problem of establishing a good theory in the still remaining range , by introducing a modified kinetic system which admits an explicit self-similar solution. The analysis of this solution clarifies the role of the exponent .     

Nearly Invariant Subspaces Related to Multiplication Operators in Hilbert Spaces of Analytic Functions

We consider Hilbert spaces of analytic functions defined on an open subset of , stable under the operator M_u of multiplication by some function u. Given a subspace of which is nearly invariant under division by u, we provide a factorization linking each element of to elements of on the inverse image under u of a certain complex disc, for which we give a relatively simple formula. By applying these results to and u(z) = z, we obtain interesting results involving a H²-norm control. In particular, we deduce a factorization for the kernel of Toeplitz operators on Dirichlet spaces. Finally, we give a localization for the problem of extraneous zeros.Submitted: January 18, 2003 Revised: December 20, 2003     

Herz Classes and Toeplitz Operators in the Disk

In this paper we decompose into diadic annuli and consider the class S_p,q of Toeplitz operators T_φ for which the sequence of Schatten norms belongs to ℓ^q, where φ_n = φχ A_n. We study the boundedness and compactness of the operators in S_p,q and we describe the operators T_φ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ.     

Hua’s theorem for five almost equal prime squares

Let p_i, 1≦ i ≦ 5, be prime numbers. It is proved that every sufficiently large integer N that satisfies N ≡ 5(mod 24) can be written as N = p²₁ + p²₂ + p²₃ + p²₄ + p²₅, where Received: 10 February 2005     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006