Calderón-Zygmund operators with kernels of Dini’s type on generalized weighted variable exponent Morrey spaces

Positivity - Let T be a Calderón-Zygmund operator of type $$omega $$ with $$omega (t)$$ being nondecreasing and satisfying a kind of Dini’s type condition and let $$T_{vec {b}}$$ be...     

Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces

In the present paper, we shall give necessary and sufficient conditions for the Spanne and Adams type boundedness of the commutators of fractional maximal operator on generalized Orlicz–Morrey spaces, respectively. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.     

Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces

The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.     

Boundedness of Calderón-Zygmund operators in product Hardy spaces

By means of vector-valued product Calderón-Zygmund operators and some subtle estimates,the boundedness in product Hardy spaces on R^n × R^m of Calderón-Zygmund operators introduced by J.L. Journé is established.     

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.     

Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces

We show that for 1 < p < ??, weight w ?? A _p , and any L ²-bounded Calderón-Zygmund operator T, there is a constant C _T,p such that the weak- and strong-type inequalities $${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$ $${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$ hold, where T _? denotes the maximal truncations of T and ${\left\| w \right\|_{{A_p}}}$ denotes the Muckenhoupt A _p characteristic of w. These estimates are not improvable in the power of ${\left\| w \right\|_{{A_p}}}$ . Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hyt?nen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T _? and sufficient conditions for two-weight inequalities in L ^p for T _?. Our proof does not rely upon extrapolation.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.     

Commutators generated by multilinear Calderón-Zygmund type singular integral and Lipschitz functions

In this paper, we establish the boundedness of commutators generated by the multilinear Calderon- Zygmud type singular integrals and Lipschitz functions on the Triebel-Lizorkin space and Lipschitz spaces.     

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Given a general dyadic grid D and a sparse family of cubes S = {Q _j ^k ∈ D, define a dyadic positive operator A _D,S by $${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$ . Given a Banach function space X(? ⁿ ) and the maximal Calderón-Zygmund operator ${T_\natural }$ , we show that $${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$ This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A ₂ conjecture”; (iii) an extension of certain mixed A _p ?A _r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A ₁ estimates (known for T ) to the maximal Calderón-Zygmund operator $\natural $ .     

Calderón-Zygmund Operators on the Predual of a Morrey Space

We consider the boundedness of Calderón-Zygmund operators from H ^{p ,ϕ} (R ⁿ )(the predual of a Morrey space) to h ^{p ,ϕ} (R ⁿ )(the local version of H ^{p ,ϕ} ). We show that Calderón's commutator is bounded from H ^{p ,ϕ} to h ^{p ,ϕ} .     

Multilinear Calderón Zygmund operators on variable exponent Morrey spaces over domains

The boundedness of multilinear singular integrals of Calderón-Zygmund type on product of variable exponent Lebesgue spaces over both bounded and unbounded domains are obtained. Further more, the boundedness for this type multilinear operators on product of variable exponent Morrey spaces over domains is shown in the paper.     

Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let(X, d, μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderón-Zygmund operator and ■:=(b1,..., bm) be a finite family of ■(μ) functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator ■Tgenerated by T and ■ are obtained.     

L~p Boundedness of Commutators of Calderón-Zygmund Singular Integral Operators

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Nonclassical weighted estimates of certain Calderónzygmund operators on the plane

Let F be a compact subset of ? let μ be a Borel measure on F, and let ρ(z) be the distance of z to F. Denote $$A_K (f)(z) = \int\limits_F {K(\varsigma ,z)f(\varsigma ) dm(\varsigma ), z \in \mathbb{C}\backslash F}$$ where K (ζ, z) is either (ζ-z)² or (|ζ-z|(ζ-z))^-1 and m is the Lebesgue measure. Let ψ be a monotone nondecreasing positive function on (0, ∞) and let Φ(z)=Ψ(ρ(z))ρ(z), z ε ?/F. Under some additional assumptions on μ, it is proved that A_K is bounded from L² (μ) to L² (Φm) if and only if $$\int\limits_0^{ + 1} {\tfrac{{\psi (t)}}{t} + \int\limits_1^\infty {\tfrac{{\psi (t)}}{{t^2 }}dt< \infty } }$$ Thus, no interference of values of K of various signs is observed in such a situation. Bibliography: 4 titles.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b∈Λβ0(Rn) is discussed from Lp(Rn) to Lq(Rn), 1/q=1/p-β0/n, and from Lp(Rn) to Triebel-Lizorkin space Fβ0,∞p. We also obtain the boundedness of generalized Toeplitz operatorθbα0 from LP(Rn) to Lq(Rn), 1/q =1/p-α0 β0/n. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on LP(Rn), 1 < p <∞.     

Boundedness of Calderón-Zygmund operators with finite non-doubling measures

Let μ be a nonnegative Radon measure on ?d which satisfies the polynomial growth condition that there exist positive constants C₀ and n ∈ (0, d] such that, for all x ∈ ?d and r>0, μ(B(x, r))≤C0rn, where B(x, r) denotes the open ball centered at x and having radius r. In this paper, we show that, if μ(?d)<∞, then the boundedness of a Calderón-Zygmund operator T on L²(μ) is equivalent to that of T from the localized atomic Hardy space h¹(μ) to L^1,∞(μ) or from h¹(μ) to L¹(μ).     

Multilinear Commutators of θ-Type Calderón-Zygmund Operators on Non-Homogeneous Metric Measure Spaces

In this paper, the boundedness in Lebesgue spaces of commutators and multilinear commutators generated by θ-type Calderón-Zygmund operators with RB MO(μ) functions on non-homogeneous metric measure spaces is obtained.