Calderon-Zygmund Singular Integrals Research Articles

On weighted Calderon-Zygmund singular integrals and applications

This paper studies some weighted norm inequalities related to some Calderon-Zygmund singular integrals. Applications to the Sobolev-Gagliardo-Nirenberg inequality, differential forms, and the potential equation du= f are given.

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderon–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces

This paper is devoted to investigating the bounded behaviors of the oscillation and variation operators for Calderon-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. We establish several criterions of boundedness, which are applied to obtain the corresponding bounds for the oscillation and variation operators of Hilbert transform, Hermitian Riesz transform and their commutators with BMO functions, or Lipschitz functions on weighted Morrey spaces.

Oscillation and variation inequalities for the commutators of singular integrals with Lipschitz functions

This paper is devoted to investigating the boundedness of the oscillation and variation operators for the commutators generated by Calderon-Zygmund singular integrals with Lipschitz functions in the weighted Lebesgue spaces and the endpoint spaces in dimension 1. Certain criterions of boundedness are given. As applications, the weighted $(L^{p}, L^{q})$ -estimates for the oscillation and variation operators on the iterated commutators of Hilbert transform and Hermitian Riesz transform, the $(L^{p}, \dot{\wedge}_{(\beta -1/p)})$ -bounds as well as the endpoint estimates for the oscillation and variation operators of the corresponding first order commutators are established.

On Regularity and Extension of Green’s Operator on Bounded Smooth Domains

We prove regularity and extension results for Green’s operators that are associated to strictly elliptic second order divergence-type linear PDO’s with coefficients in \(C^{1,\alpha}(\overline{\Omega})\). Here α ∈ (0, 1) and Ω ⊂ R n , n ≥ 3, is a bounded C 2,α domain. The regularity result gives boundary estimates for the derivatives up to order (2 + α) of the associated Green’s function. With the aid of this regularity result, we then extend the Green’s operator to a globally defined integral operator whose second order partial derivatives are Calderon–Zygmund singular integrals. We also show that, under reasonable a priori assumptions, the C 2,α regularity of the domain is necessary for the aforementioned extension of the Green’s operator to a weakly singular integral operator, belonging to the class \({\rm{SK}}^{-2}_{{\bf{R}}^n}(\alpha)\).

New Estimates for the Maximal Singular Integral

In this paper, we pursue the study of the problem of controlling the maximal singular integral T * f by the singular integral Tf. Here T is a smooth homogeneous Calderon-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the norm and via pointwise estimates of T * f by M(Tf) or M 2 (Tf), where M is the Hardy-Littlewood maximal operator and its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article, we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak (1,1) estimates, which are no longer true for odd kernels. For odd kernels, we obtain sharp weaker inequalities involving a weak L 1 estimate for functions in L log L.

The foundational inequalities of D. L. Burkholder and some of their ramifications

This paper presents an overview of some of the applications of the martingale inequalities of D. L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling–Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calderon–Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in the norms, hence our restriction to the above operators. Connections of Burkholder’s foundational work on sharp martingale inequalities to other areas of mathematics where either the results themselves or techniques to prove them have become of considerable interest in recent years, are discussed. These include the 1952 conjecture of C. B. Morrey on rank-one convex and quasiconvex functions with connections to problems in the calculus of variations and the 1982 conjecture of T. Iwaniec on the $L^p$-norm of the Beurling–Ahlfors operator with connections to problems in the theory of qasiconformal mappings. Open questions, problems and conjectures are listed throughout the paper and copious references are provided.

A new approach to Lp estimates for Calderón-Zygmund type singular integrals

In this paper we give a new proof for L p estimates of the Calderon-Zygmund type singular integrals. Our approach is completely free from harmonic analysis.

Fractional Integral Operators on Anisotropic Hardy Spaces

In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and \(m \in {\mathbb{N}}\). First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H p to H q , or from H p to L q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderon-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function.

On some weighted norm inequalities for Littlewood–Paley operators

It is shown that the $L^p_w,1 2$. As a corollary, a new estimate in terms of $\|w\|_{A_p}$ is obtained for the class of Calderon-Zygmund singular integrals commuting with dilations.

The Hardy–Lorentz spaces Hp,q(Rn)

We deal with the Hardy-Lorentz spaces H p,q (R n ) where 0 < p � 1, 0 < q � 1. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them. The real variable theory of the Hardy spaces represents a fruitful set- ting for the study of maximal functions and singular integral operators. In fact, it is because of the failure of these operators to preserve L 1 that the Hardy space H 1 assumes its prominent place in harmonic analysis. Now, for many of these operators, the role of L 1 can just as well be played by H 1,∞ , or Weak H 1 . However, although these operators are amenable to H 1 -L 1 and H 1,∞ -L 1,∞ estimates, interpolation between H 1 and H 1,∞ has not been available. Similar considerations apply to H p and Weak H p for 0 < p < 1. The purpose of this paper is to provide an interpolation result for the Hardy-Lorentz spaces H p,q , 0 < p ≤ 1, 0 < q ≤ ∞, including the case of Weak H p as an end point for real interpolation. Since in this context neither truncations are available nor reiteration applies, the atomic decomposition will be the key ingredient in dealing with interpolation. The paper is organized as follows. The Lorentz spaces, including criteria that ensure membership in L p,q , 0 < p < ∞, 0 < q ≤ ∞, are discussed in Section 1. In Section 2 we show that distributions in H p,q have an atomic de- composition in terms of H p atoms with coefficients in an appropriate mixed norm space. An interesting application of this decomposition is to H p,q -L p,∞ estimates for Calderon-Zygmund singular integral operators, p < q ≤ ∞. Also, by manipulating the different levels of the atomic decomposition, we show that, for 0 < q1 < q < q2 ≤ ∞, H p,q is an intermediate space between H p,q1 and H p,q2 . This result applies to Calderon-Zygmund singular integral

Maximal calderón-zygmund singular integral on RBMO

Given a positive Radon measure μ on Rd satisfying the linear growth condition μ(B(x,r))≤C0rn,x∈Rd,r>0, (1) where n is a fixed number and 0<n≤d. When d−1<n, it is proved that if T∈,N1=0, then the corresponding maximal Calderon-Zygmund singular integral is bounded from RBMO to itself only except that it is infinite μ-a.e. on Rd.

Weighted norm inequalities for maximally modulated singular integral operators

We present a framework that yields a variety of weighted and vector-valued estimates for maximally modulated Calderon-Zygmund singular (and maximal singular) integrals from a single a priori weak type unweighted estimate for the maximal modulations of such operators. We discuss two approaches, one based on the good-λ method of Coifman and Fefferman [CF] and an alternative method employing the sharp maximal operator. As an application we obtain new weighted and vector-valued inequalities for the Carleson operator proving that it is controlled by a natural maximal function associated with the Orlicz space L(log L)(log log log L). This control is in the sense of a good-λ inequality and yields strong and weak type estimates as well as vector-valued and weighted estimates for the operator in question.

An endpoint estimate for the commutators of singular integrals with non doubling measures

A weak type endpoint estimate is established for commutators of the Calderon-Zygmund singular integrals onRd with non doubling measures.

Maximal operator and weighted norm inequalities for multilinear singular integrals

The maximal operator associated with multilinear Calderon-Zygmund singular integrals is introduced and shown to be bounded on product of Lebesgue spaces. Moreover weighted norm inequalities are obtained for this maximal operator as well as for the corresponding singular integrals.

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝ n are bounded from $$ H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha _{1} ,p_{1} }}_{{q1}} {\left( {\mathbb{R}^{n} } \right)}\times \cdots \times{\kern 1pt} H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha _{k} ,p_{k} }}_{{q_{k} }} {\left( {\mathbb{R}^{n} } \right)} $$ into $$ H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha ,p}}_{q} {\left( {\mathbb{R}^{n} } \right)} $$ if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when αj≥ 0 and the singular integrals considered here include the Calderon-Zygmund singular integrals and the fractional integrals of any orders.

A rearranged good 𝜆 inequality

Let T f Tf be a maximal Calderón-Zygmund singular integral, M f Mf the Hardy-Littlewood maximal function, and w w an A ∞ {A_\infty } weight. We replace the “good λ \lambda ” inequality \[ w ( { x : T f ( x ) &gt; 2 λ and M f ( x ) ≤ ε λ } ) ≤ C ( ε ) w ( { x : T f ( x ) &gt; λ } ) w\left ( {\{ x:\,Tf(x) &gt; 2\lambda \,{\text {and}}\,Mf(x) \leq \varepsilon \lambda \} } \right ) \leq C(\varepsilon )w\left ( {\{ x:\,Tf(x) &gt; \lambda \} } \right ) \] by the rearrangement inequality \[ ( T f ) w ∗ ( t ) ≤ C ( M f ) w ∗ ( t / 2 ) + ( T f ) w ∗ ( 2 t ) (Tf)_w^ \ast (t) \leq C(Mf)_w^ \ast (t/2) + (Tf)_w^ \ast (2t) \] and show that it gives better estimates for T f Tf . In particular, we obtain best possible weighted L p {L^p} bounds, previously unknown exponential integrability estimates, and simplified derivations of known unweighted estimates for ( T f ) ∗ {(Tf)^ \ast } .

On a problem of S. G. Mikhlin concerning the behavior of singular integrals at infinity

A solution is given of a problem formulated by S. G. Mikhlin concerning the behavior of Calderon-Zygmund singular integrals at infinity.