Hausdorff Operators on Function Spaces

The authors mainly study the Hausdorff operators on Euclidean space Rn.They establish boundedness of the Hausdorff operators in various function spaces,such as Lebesgue spaces,Hardy spaces,local Hardy ...     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

Criteria for the compactness and the Fredholm property of pseudodifferential operators in Hölder-Zygmund weighted spaces

We obtain necessary and sufficient conditions for the complete continuity (the Fredholm property) in Hölder-Zygmund spaces on ? ⁿ whose weight has a power-law behavior at infinity for pseudodifferential operators with symbols in the Hörmander class S _1,δ ^m , 0 ≤ δ < 1 (slowly varying symbols in the class S _1,0 ^m ). We show that such operators are compact operators or Fredholm operators in weighted Hölder-Zygmund spaces if and only if they are compact operators or Fredholm operators, respectively, in Sobolev spaces.     

Orlicz-Hardy spaces and their duals

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H(Rn)in that the class of admissible functions is largely widened.We can deal with,for example,(r)≡(rp1(log(e+1/r))q1,0r 6 1,rp2(log(e+r))q2,r1,with p1,p2∈(0,∞)and q1,q2∈(.∞,∞),where we shall establish the boundedness of the Riesz transforms on H(Rn).In particular,is neither convex nor concave when 0p11p2∞,0p21p1∞or p1=p2=1 and q1,q20.If(r)≡r(log(e+r))q,then H(Rn)=H(logH)q(Rn).We shall also establish the boundedness of the fractional integral operators I of order∈(0,∞).For example,I is shown to be bounded from H(logH)1./n(Rn)to Ln/(n.)(log L)(Rn)for 0n.     

Anisotropic singular integrals in product spaces

In this paper, the authors introduce a class of product anisotropic singular integral operators, whose kernels are adapted to the action of a pair A := (A1, A2) of expansive dilations on R n and R m , respectively. This class is a generalization of product singular integrals with convolution kernels introduced in the isotropic setting by Fefferman and Stein. The authors establish the boundedness of these operators in weighted Lebesgue and Hardy spaces with weights in product A∞ Muckenhoupt weights on R n × R m . These results are new even in the unweighted setting for product anisotropic Hardy spaces.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

Characterization of pseudodifferential operators in Hölder-Zygmund spaces

We consider pseudodifferential operators with symbols of the Hörmander class S _{1, δ} ^m , 0 ≤ δ < 1, in Hölder-Zygmund spaces on ? ⁿ and obtain a Beals-type characterization of such operators. By way of application, we show that the inverse of a pseudodifferential operator invertible in a Hölder-Zygmund space is itself a pseudodifferential operator, and hence, the spectra of a pseudodifferential operator in the space L ₂ and in the Hölder-Zygmund spaces coincide as sets.     

Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.     

Finite Dimensional de Branges Spaces on Riemann Surfaces

We study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators Rα; here we introduce generalized difference quotient operators Ryα for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators Ry1αi and Ry2α2, where y1 and y2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks.     

Fredholmness of pseudodifference operators in weighted Spaces

The aim of the paper is to present some results concerning pseudodifference operators on ?^N, which are a discrete analog of standard pseudodifferential operators on ?^N. We study the Fredholm property of pseudodifference operators acting in weighted l ^p spaces on ?^N and the Phragmen-Lindelöf principle for solutions of pseudodifference equations and give applications of these results to discrete Schrö dinger operators on ?^N.     

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.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

Tent spaces associated with semigroups of operators

We study tent spaces on general measure spaces (Ω,μ). We assume that there exists a semigroup of positive operators on Lp(Ω,μ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's H¹-BMO duality theory. We also get a H¹-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative Lp spaces.     

Construction of Pathological Maximally Monotone Operators on Non-reflexive Banach Spaces

In this paper, we construct maximally monotone operators that are not of Gossez’s dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC–functions will not always be a BC–function. This provides a negative answer to a challenging question posed by Stephen Simons. Among other consequences, we deduce—in a uniform fashion—that every Banach space which contains an isomorphic copy of the James space \({\ensuremath{\mathbf{J}}}\) or its dual \({\ensuremath{\mathbf{J}}}^{\ast}\), or c ₀ or its dual ?¹, admits a non type (D) operator. The existence of non type (D) operators in spaces containing ?¹ or c ₀ has been proved recently by Bueno and Svaiter.     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

An Inverse Function Theorem for Fréchet Spaces Satisfying a Smoothing Property and (DN)

Classical inverse function theorems of Nash-Moser type are proved for Fréchet spaces that admit smoothing operators as introduced by Nash. In this note an inverse function theorem is proved for Fréchet spaces which only have to satisfy the condition (DN) of Vogt and the smoothing property (S_Ω)_t; for instance, any Fréchet-Hilbert space which is an (Ω)-space in standard form has property (S_Ω)_t. The main result of this paper generalizes a theorem of Lojasiewicz and Zehnder. It can be applied to the space C^∞(K) if the compact K ? ?^N is the closure of its interior and subanalytic; different from classical results the boundary of K may have singularities like cusps. The growth assumptions on the mappings are formulated in terms of the weighted multiseminorms [ ]m,k introduced in this paper; nonlinear smooth partial differential operators on C^∞(K) and their derivatives satisfy these formal assumptions.     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

Cotype and absolutely summing linear operators

New applications of cotype to the theory of absolutely summing linear operators between Banach spaces are proved in this paper. Among other consequences we extend/complement some classical results of Bennett (Duke Math J 44:603?C639, 1977) on the existence of non-absolutely summing operators between ? _p spaces and of Davis and Johnson (Stud Math 51:81?C85, 1974) on the existence of compact non-absolutely summing linear operators. We also point out that some of our results are sharp.     

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.