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 $ .     

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 <∞.     

Endpoint estimates for Calderón-Zygmund type operators

In this paper, the author states three classes of endpoint estimates for Calderon-Zygmund type operators. As their special cases, the corresponding results of classical Calderon-Zygmund operators can be deduced respectively.     

Θ-type Calderón-Zygmund operators with non-doubling measures

Let µ be a Radon measure on ? ^d which may be non-doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr ⁿ for all x∈? ^d , r > 0 and for some fixed 0 < n ≤ d. In this paper, under this assumption, we prove that θ-type Calderón-Zygmund operator which is bounded on L ²(µ) is also bounded from L ^∞(µ) into RBMO(µ) and from H _atb ^1,∞ (µ) into L ¹(µ). According to the interpolation theorem introduced by Tolsa, the L ^p (µ)-boundedness (1 < p < ∞) is established for θ-type Calderón-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type Calderón-Zygmund operator with RBMO(µ) function are bounded on L ^p (µ) (1 < p < ∞).     

Maximal multilinear Calderón-Zygmund operators with non-doubling measures

Under the assumption that μ is a non-doubling measure on ℝ ^d which only satisfies some growth condition, the authors prove that the maximal multilinear Calderón-Zygmund operator is bounded from (μ) × … × (μ) into L ^p (μ) for any p ₁, … p _m ∈ (1, ∞) and p with 1/p = 1/p ₁ + … + 1/p _m , and bounded from (μ) × … × (μ) into weak- ^{L p} (μ) if there exists any p _i = 1. Furthermore, the authors establish a weighted weak-type estimate for the maximal multilinear Calderón-Zygmund operator. The second (corresponding) author is supported by Tianyuan Fund of China (No. 10626055) and National Natural Science Foundation of China (No. 10701078).     

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.     

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).     

A necessary condition for Calderón-Zygmund singular integral operators

Calderón-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderón-Zygmund convolution singular integral operator is bounded on the Hardy space H¹ (Rⁿ), then the homogeneous of degree zero kernel is in the Hardy space H¹(S^n–1) on the sphere.     

Calderón–Zygmund operators in the Bessel setting

We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood–Paley–Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderón–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.     

Weighted norm inequalities for the multilinear Calderón-Zygmund operators

In this paper, by proving some suitable weighted endpoint estimates, and then by multilinear interpolation and a new multilinear extrapolation lemma, the author establishes some weighted estimates for the multilinear Calderón-Zygmund operator. Also, the author gives a weighted estimate for the corresponding commutator.     

The Calderón problem for variable coefficients nonlocal elliptic operators

In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator (???(A(x)?))^s+q), for 0<s<1. We determine the unknown bounded potential q from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension n≥2. Our results generalize the recent initiative [18 Ghosh, T., Salo, M., Uhlmann, G. (2016). The Calderón problem for the fractional Schrödinger equation. arXiv preprint arXiv:1609.09248. [Google Scholar]] of introducing and solving inverse problem for fractional Schrödinger operator ((?Δ)^s+q) for 0<s<1. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.     

Wavelet shrinkage estimators of Calderón-Zygmund operators with odd kernels

Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given function f and is widely used in data compression,signal processing and statistics,etc.For Calder′on-Zygmund operators T,it is interesting to construct estimator of T f,based on wavelet shrinkage estimator of f.With the help of a representation of operators on wavelets,due to Beylkin et al.,an estimator of T f is presented in this paper.The almost everywhere convergence and norm convergence of the proposed estimators are established.     

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.     

On Some Geometric Properties of Generalized Calderón-Lozanovskii Spaces

Strict monotonicity, lower and upper local uniform monotonicities, rotundity, uniform rotundity and uniform non-squareness of generalized Calderón-Lozanovskii spaces are studied.     

On local oscillatory integrals with variable Calderón-Zygmund kernels

Letp(1, ). In this paper, the authors investigate the uniformL ^p ( ⁿ ) in of the oscillatory singular integral operatorT defined by