The ρ-variation of the Hermitian Riesz transform

In this paper we prove the behaviour in weighted Lp spaces of the oscillation and variation of the Hilbert transform and the Riesz transform associated with the Hermite operator of dimension 1. We prove that this operator maps LP（R, w（x）dx） into itself when w is a weight in the Ap class for 1 〈 p 〈 ∞. For p = 1 we get weak type for the A1 class. Weighted estimated are also obtained in the extreme case p = ∞.     

Strongly singular convolution operators on the weighted Herz-type Hardy spaces

Let 0<p≤1<q<0, andw ₁ ,w ₂ ∈ A ₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spacesH K^{α, p} _q(w₁; w₂) to the homogeneous weighted Herz spacesK ^{α, p} _q (w₁; w₂), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K ^{α, p} _q (w ₁;w ₂) is also investigated. Supported by the National Natural Science Foundation of China     

Boundedness and Fredholmness of Pseudodifferential Operators in Variable Exponent Spaces

We prove a statement on the boundedness of a certain class of singular type operators in the weighted spaces with variable exponent p(x) and a power type weight w, from which we derive the boundedness of pseudodifferential operators of H?rmander class S ⁰ _1,0 in such spaces. This gives us a possibility to obtain a necessary and sufficient condition for pseudodifferential operators of the class OPS ^m _1,0 with symbols slowly oscillating at infinity, to be Fredholm within the frameworks of weighted Sobolev spaces with constant smoothness s, variable p(·)-exponent, and exponential weights w. Supported by CONACYT Project No.43432 (Mexico), the Project HAOTA of CEMAT at Instituto Superior Técnico, Lisbon (Portugal) and the INTAS Project “Variable Exponent Analysis” Nr.06-1000017-8792.     

Local Hardy–Littlewood maximal operator

In this article we define and investigate a local Hardy–Littlewood maximal operator in Euclidean spaces. It is proved that this operator satisfies weighted L ^p , p > 1, and weighted weak type (1,1) estimates with weight function ${w \in A^p_{\rm{loc}}}In this article we define and investigate a local Hardy–Littlewood maximal operator in Euclidean spaces. It is proved that this operator satisfies weighted L ^p , p > 1, and weighted weak type (1,1) estimates with weight function w ? A^p_loc{w \in A^p_{\rm{loc}}}, the class of local A _p weights which is larger than the Muckenhoupt A _p class. Also, the condition w ? A^p_loc{w \in A^p_{\rm{loc}}} turns out to be necessary for the weighted weak type (p,p), p ≥ 1, inequality to hold.     

Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type

We determine the exact asymptotic order of the entropy numbers of compact embeddings of weighted Besov spaces in the case where the ratio of the weights w(x) = w ₁(x)/w ₂(x) is of logarithmic type. This complements the known results for weights of polynomial type. The estimates are given in terms of the number 1/p = 1/p ₁ − 1/p ₂ and the function w(x). We find an interesting new effect: if the growth rate at infinity of w(x) is below a certain critical bound, then the entropy numbers depend only on w(x) and no longer on the parameters of the two Besov spaces. All results remain valid for Triebel–Lizorkin spaces as well.     

Boundedness criteria for singular integrals in weighted grand lebesgue spaces

Boundedness criteria for the Calderón singular integral, Riesz transform and Cauchy singular integral in generalized weighted grand Lebesgue spaces L ^p),θ _w , 1 < p < ∞, are studied. It is shown that an operator K of this type is bounded in L ^p),θ _w if and only if the weight w satisfies the Muckenhoupt A _p condition. Bibliography: 15 titles.     

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

Boundedness of Marcinkiewicz integral on the weighted Herz-type Hardy spaces

In this paper, we discuss the boundedness of Marcinkiewicz integral μΩ with homogeneous kernel on the weighted Herz-type Hardy spaces, and prove that μΩ is bounded from H K αq ,p ( w1 ; w2 ) into Kαq ,p (w 1; w2).     

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.     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

Removable singularities for weighted Bergman spaces

We develop a theory of removable singularities for the weighted Bergman space , where μ is a Radon measure on ℂ. The set A is weakly removable for , and strongly removable for . The general theory developed is in many ways similar to the theory of removable singularities for Hardy H ^p spaces, BMO and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if μ is absolutely continuous with respect to the Lebesgue measure m, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When dμ = wdm and w is a Muckenhoupt A _p weight, 1 < p < ∞, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent p′ = p/(p − 1) and the dual weight w′ = w ^{1/(1 − p)}.     

A Schur test for weighted mixed-norm spaces

Summary We prove a Schur test for mixed-norm spaces L^p,q, 1 < p,q < ∞. Also we prove another version of the Schur test for discrete weighted mixed-norm spaces l^p,q _w, 1 < p,q < ∞, and wis a weight. We show that if w ₁, and w ₂are two weight functions on the index sets Jx Iand K x Lrespectively, and A =(a _{ji, kl} ) _{j∈J, i∈I, k∈K, l∈L} is an infinite matrix, then under certain conditions, Ais a bounded operator from l^p,q _w1, 1 < p,q < ∞ to l^p,q _w2. This will be a key result in proving boundedness of important operators in our work in time-frequency analysis.</o:p>     

The Calderón-Zygmund decomposition and interpolation on weighted Hardy spaces

We apply discrete Littlewood-Paley-Stein theory, developed by Han and Lu, to establish Calderón-Zygmund decompositions and interpolation theorems on weighted Hardy spaces H _w ^p for ω ∈ A _∞ in both the one-parameter and two-parameter cases.     

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

Spectra of weighted composition operators on weighted banach spaces of analytic functions

We determine the spectra of weighted composition operators acting on the weighted Banach spaces of analytic functionsH _{ν p} ^∞ when the symbolφ has a fixed point in the open unit disk. Further, we apply this result to give the spectra of composition operators on Bloch type spaces. In particular, we answer in the affirmative a conjecture by MacCluer and Saxe. The research of the second author was partially supported by the Academy of Finland Project No. 51906; the research of this paper was carried out while this author was visiting Kent State University, whose hospitality is acknowledged with thanks.     

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ^??(w) in terms of the fundamental function of L ^??(w). In particular, we prove that these bases are greedy in L ^??(w) if and only if L ^??(w) =?L ^p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L ^p (w) the approximation spaces are identified with weighted Besov spaces.     

Spaces of functions of fractional smoothness on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce spaces of functions of fractional smoothness s > 0. We prove embedding theorems relating these spaces to the Sobolev spaces W _p ^m (G) and Lebesgue spaces L _p (G).     

On the Representation of Compact Nilpotent Rings by Triangular Matrices

We prove that every compact nilpotent ring R of characteristic p > 0 can be embedded in a ring of upper triangular matrices over a compact commutative ring. Furthermore, we prove that every compact topologically nilpotent ring R of characteristic p > 0, is embedded in a ring of infinite triangular matrices over \mathbbF_p^w(R)\mathbb{F}_{p}^{w(R)}.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces L_w²(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? C_c(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L²_w(\mathbb R^k){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.