A Note on Classes of BANACH Spaces Related to Stable Measures

The problem under consideration is the following: Let S: E′ → L_q, T: E′ → L_p, 0 < q ≦ 2, 0 < p ≦ 2, be operators, ‖Sa‖ ≦ ‖Ta‖, such that, T generates a stable measure on E, i.e., exp (-‖Ta‖ ^p), a ? E′, is the characteristic function of a RADON measure on E. Does this imply, that exp (-‖Sa‖ q), a ? E′, is the characteristic function of a RADON measure, too? In general this is not true provided q or p less than 2. A BANACH space is said to be of (q,p)-cotype if the answer to the above question is “yes”. We establish several properties of this classification and obtain as an application the well-known classes due to MOUCHTARI, TIEN, WERON and MANDREKAR, WERON, Finally we apply our results to so-called S-spaces.     

Banach space sequences and projective tensor products

In this paper we use results from the theory of tensor products of Banach spaces to establish the isometry of the space of (1,p)-summing sequences (also known as strongly p-summable sequences) in a Banach space X, the space of nuclear X-valued operators on ?^p and the complete projective tensor product of ?^p with X. Through similar techniques from the theory of tensor products, the isometry of the sequence space L^p〈X〉 (recently introduced in a paper by Bu, Quaestiones Math. (2002), to appear), the space of nuclear X-valued operators on L^p(0,1) (with a suitable equivalent norm) and the complete projective tensor product of L^p(0,1) with X is established. Moreover, we find conditions for the space of (p,q)-summing multipliers to have the GAK-property (generalized AK-property), use multiplier sequences to characterize Banach space valued bounded linear operators on the vector sequence space of absolutely p-summable sequences in a Banach space and present short proofs for results on p-summing multipliers.     

On boundedness of the hardy and bellman operators in the spaces h and bmo

Hardy (1919) proved that the space L^p(T), 1 ? p ? ∞, is invariant under (C, 1)- transformation of Fourier coefficients. This transformation may be considered as a linear integral operator H: L^p(T) → L^p(T), 1 ? p ? ∞. In this paper we show that the operator H is unbounded in the space BMO(T) of periodic functions of bounded mean oscillation. For the conjugate operator B introduced by Bellman (1944) B: L^q(T) → L^q(T), 1/ρp + 1/rho;q = 1, the boundedness of B in BMO(T) is proved directly without duality argument. An example is given to show that B is unbounded in H(T).     

Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some results concerning Lp(μ)-spaces

It is proved that L^p(μ) does not have an unconditional basis if the cardinality of L^p(μ) is sufficiently large and μ is a finite measure. It is also shown that L^p(μ) has a weaker kind of basis for arbitrary μ and 1 < p < ∞. A new truncation lemma concerning sequences in L^p equivalent to the usual l^p-basis is given. This lemma is used in solving the problem of when l^p(Γ) imbeds in L^r(μ) for uncountable sets Γ and finite measures μ. It may also be used to give a nonprobabilistic proof of the fact (due to Schwartz-Kwapien) that there exist non-q-absolutely summing operators from L^∞to L^qfor 2 < q < ∞. It is again used in proving that basic sequences in L^p equivalent to the usual l^p-basis admit subsequences with a complemented linear span. Other applications of the techniques introduced are also given.     

On a Hardy Type General Weighted Inequality in Spaces L p(·)

A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L ^p(·). Equivalent necessary and sufficient conditions are found for the ${L^{p(\cdot)} \longrightarrow L^{q(\cdot)}}A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L ^p(·). Equivalent necessary and sufficient conditions are found for the L^p(·) ? L^q(·){L^{p(\cdot)} \longrightarrow L^{q(\cdot)}} boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.     

Two-point inequalities,the Hermite semigroup and the Gauss-Weierstrass semigroup

Let e^?zH, Re z ? 0, be the Hermite semigroup on R with Gauss measure μ. Necessary and sufficient conditions for e^?zH to be a bounded map from L^p(μ) into L^q(μ), 1 ? p, q ? ∞, are found and in many cases it is proved that e^?zH: L^p(μ) → L^q(μ) is in fact a contraction. Furthermore, these results and a formula relating the Hermite semigroup with the Gauss-Weierstrass semigroup e^zΔ enable one to calculate the precise norm of e^zΔ:L^p(dx) → L^q(dx) in a large number of cases.     

Criteria of Nuclearity of Embedding Operators Acting between BERGMAN Spaces and Weighted LEBESQUE Spaces

In the paper the criteria of nuclearity of embedding operators acting from BERGMAN space into the space L_q(μ) are established. Some questions related to P_r – summability of these operators are considered. The criteria of nuclearity of embedding operator from H^p into L_q(μ) for some class of measures and p ≧ q are given.     

Potential operators and laplace type multipliers associated with the twisted laplacian

We study potential operators and,more generally,Laplace-Stieltjes and Laplace type multipliers associated with the twisted Laplacian.We characterize those 1 ≤ p,q ≤∞,for which the potential operators are L~p—L~q bounded.This result is a sharp analogue of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the context of special Hermite expansions.We also investigate L~p mapping properties of the Laplace-Stieltjes and Laplace type multipliers.     

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗_q on certain quasi-Banach spaces. We prove that the approximation spaces A^α_q(L₂) and A^α_q(H¹) equal tensor products of Besov spaces B^α_q(L_q), e.g., A^α_q(L₂([0,1]^d)) = B^α_q(L_q([0,1])) ⊗_q · ⊗_q B^α_{q · ·}(L_q([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.     

Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.     

AN EIGENSPACE CHARACTERIZATION OF LORENTZ-IMPROVING MEASURES ON COMPACT GROUPS

Abstract

A measure μ on a compact group is called Lorentz-improving if for some 1 > p > ∞ and 1 → q ₁ > q ₂ ∞ μ *L (p, q ₂) ? L(p, q ₁). Let T _μ denote the operator on L ² defined by T _μ(f) = μ * f. Lorentz-improving measures are characterized in terms of the eigenspaces of T _μ, if T _μ is a normal operator, and in terms of the eigenspaces of |T _μ| otherwise. This result generalizes our recent characterization of Lorentz-improving measures on compact abelian groups and is modelled after Hare's characterization of L ^p -improving measures on compact groups.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

The annihilator of Bergman space

LetD be a bounded plane domain (with some smoothness requirements on its boundary). LetB _p(D), 1≤p<∞, be the Bergmanp-space ofD. In a previous paper we showed that the “natural projection”P, involving the Bergman kernel forD, is a bounded projection fromL _p(D) ontoB _p(D), 1<p<∞. With this we have the decompositionL _p(D)=B _p(D)⊕B _q ^⊥ (D,p ^–1+q ^–=1, 1<p< ∞. Here, we show that the annihilatorB _q ^⊥ (D) is the space of allL _p-complex derivatives of functions belonging to Sobolev space and which vanish on the boundary ofD. This extends a result of Schiffer for the casep=2. We also study certain operators onL _p(D). Especially, we show that , whereI is the identity operator and ? is an operator involving the adjoint of the Bergman kernel. Other relationships relevant toB _q ^⊥ (D) are studied.     

Boundedness of Schrödinger Type Propagators on Modulation Spaces

It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces ? ^p,q , for 1≤p= q≤∞, provided their symbols belong to the Sjöstrand class M ^∞,1. However, they generally fail to be bounded on ? ^p,q for p≠q. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on ? ^p,q for p≠q, and between ? ^p,q →? ^q,p , 1≤q<p≤∞. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.     

Lp-Computability

In this paper we investigate conditions for L^p-computability which are in accordance with the classical Grzegorczyk notion of computability for a continuous function. For a given computable real number p ≥ 1 and a compact computable rectangle I ? ?^q, we show that an L^p function f ∈ L^p(I) is L^P-computable if and only if (i) f is sequentially computable as a linear functional and (ii) the L^p-modulus function of f is effectively continuous at the origin of ?^q.     

τ-可测正算子生成的交换子的若干不等式

设M是具有正规忠实的半有限迹τ的von Neumann代数,‖.‖ρ是任意非交换Banach函数空间范数,‖.‖是M上的通常范数.证明了若A和B是τ-可测正算子,X∈M,则‖AX-XB‖ρ≤‖X‖‖AB‖ρ.还证明了若A,B是M中的正算子,X是τ-可测算子,则‖AX-XB‖ρ≤max(‖A‖,‖B‖)‖X‖ρ.由此得到了若A∈M是正算子,X是τ-可测正算子,则‖AX-XA‖ρ≤1/2‖A‖‖XX‖ρ.     

BOUNDED LINEAR OPERATORS THAT COMMUTE WITH SHIFTS ARE SCALED IDENTITY

We prove that every bounded linear operator on L^p?L^p(R'),s≥1 and 1≤p<∞, that commutes with both the spatial shift operator and the phase shift operator must be a constant multiple of the identity. We also apply this result to identify analysis-synthesis dual L^q-L^p paris and to characterize bi-orthogonal unconditional bases of L^q-L^p, where p^?1+q^?1=1.     

A note on the sunouchi operator with respect to the walsh—kaczmarz system

An analogue of the so—called Sunouchi operator with respect to the Walsh—Kaczmarz system will be investigated. We show the boundedness of this operator if we take it as a map from the dyadic Hardy space H ^p to L ^p for all 0<p≤1.. For the proof we consider a multiplier operator and prove its (H ^p H ^p)—boundedness for 0<p≤1. Since the multiplier is obviously bounded from L ² to L ², a theorem on interpolation of operators can be applied to show that our multiplier is of weak type (1,1) and of type (q q) for all 1<q<∞. The same statements follow also for the Sunouchi operator.