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.     

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.     

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 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.     

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.     

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).     

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.