The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces

Operator Theory: Advances and Applications, 2013

Let p : R → (1, ∞) be a globally log-Hölder continuous variable exponent and w : R → [0, ∞] be a weight. We prove that the Cauchy singular integral operator S is bounded on the weighted variable Lebesgue space L p(•) (R, w) = {f : f w ∈ L p(•) (R)} if and only if the weight w satisfies sup −∞<a<b<∞ 1 b − a wχ (a,b) p(•) w −1 χ (a,b) p ′ (•) < ∞ (1/p(x) + 1/p ′ (x) = 1).

Authorea

The paper deals with boundedness problems of integral operators in weighted grand Bochner-Lebesgue spaces. We will treat both cases: when a weight function appears as a multiplier in the definition of the norm, or when it defines the absolute continuous measure of integration. Along with the diagonal case, we deal with the off-diagonal case. To get the appropriate result for the Hardy-Littlewood maximal operator, we rely on the reasonable bound of the sharp constant in the Buckley-type theorem, which is also derived in the paper.

arXiv (Cornell University), 2010

Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.

Taiwanese Journal of Mathematics, 2012

In this paper we prove the boundedness of certain convolution operator in a weighted Lebesgue space with kernel satisfying the generalized Hörmander's condition. The sufficient conditions for the pair of weights ensuring the validity of two-weight inequalities of a strong type and of a weak type for singular integral with kernel satisfying the generalized Hörmander's condition are found.

Bulletin des Sciences Mathématiques, 2013

Let T be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of R n. We assume that the associated kernel of T satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for T and the boundedness of the commutator of T with BMO functions on products of weighted Lebesgue spaces of R n. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of their commutators with BMO functions on the products of weighted Lebesgue spaces when the number of derivatives of the symbols is the same as the best known result for the multilinear Fourier multipliers to be bounded on the products of unweighted Lebesgue spaces. Contents

Analysis &amp; PDE, 2012

Let σ and ω be positive Borel measures on ‫ޒ‬ with σ doubling. Suppose first that 1 < p ≤ 2. We characterize boundedness of certain maximal truncations of the Hilbert transform T from L p (σ) to L p (ω) in terms of the strengthened A p condition ‫ޒ‬ s Q (x) p dω(x) 1/ p ‫ޒ‬ s Q (x) p dσ (x) 1/ p ≤ C|Q|, where s Q (x) = |Q|/(|Q| + |x − x Q |), and two testing conditions. The first applies to a restricted class of functions and is a strong-type testing condition, Q T (χ E σ)(x) p dω(x) ≤ C 1 Q dσ (x) for all E ⊂ Q, and the second is a weak-type or dual interval testing condition, Q T (χ Q f σ)(x) dω(x) ≤ C 2 Q | f (x)| p dσ (x) 1/ p Q dω(x) 1/ p for all intervals Q in ‫ޒ‬ and all functions f ∈ L p (σ). In the case p > 2 the same result holds if we include an additional necessary condition, the Poisson condition ‫ޒ‬ ∞ r =1 |I r | σ |I r | p −1 ∞ =0 2 − |(I r) () |χ (I r) () (y) p dω(y) ≤ C ∞ r =1 |I r | σ |I r | p , for all pairwise disjoint decompositions Q = ∞ r =1 I r of the dyadic interval Q into dyadic intervals I r. We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in ‫ޒ‬ n when σ is doubling and 1 < p < ∞. Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals T in ‫ޒ‬ n when 1 < p < ∞, without the doubling assumption on σ , in terms of analogues of the second testing condition and the A p condition.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2011

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al. This class of operators extends the class of multi-linear Calderón-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar&#39;s inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced.

Mathematische Annalen, 2004

We present a framework that yields a variety of weighted and vector-valued estimates for maximally modulated Calderón-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 and also for a related maximally modulated operator with quadratic phase studied by M. Lacey. We obtain that these operators are 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 operators in question. T f (x) = R n K(x, y) f (y) dy, a.e. x ∈ R n \ supp f. The kernel K : R n × R n \ {(x, x) : x ∈ R n } −→ C is assumed to satisfy the following standard conditions |K(x, y)| ≤ c 0 |x − y| n , x = y, and |K(x, y) − K(x, y)| + |K(y, x) − K(y , x)| ≤ c 0 |y − y | τ |x − y| n+τ , |x − y| > 2 |y − y |, for some c 0 , τ > 0. Associated with T there is a truncated operator T ε and a maximal singular operator T defined as follows: T ε f (x) = |x−y|>ε K(x, y) f (y) dy, T f (x) = sup ε>0 |T ε f (x)|.

Filomat

A number of properties for the classes Bp-1 and B*p have been proved. The class Bp-1 characterizes the Lp- inequality involving the averaging operator and the class B*p characterizes the Lp-inequality involving the adjoint averaging operator. The reverse inequalities involving the integral operators in Lp? have also been studied.

1999

We give a sufficient condition for singular integral operators and, more generally, Calder´on-Zygmund operators to satisfy the weak (p, p) inequality u({x ∈ R n : |T f(x)| &gt; t}) ≤ C / tp Z Rn |f|p v dx, 1 &lt; p &lt; ∞. Our condition is an Ap-type condition in the scale of Orlicz spaces: kukL(log L) p−1+δ,Q 1 |Q| Z Q v −p 0/p dx p/p0 ≤ K 0. This conditions is stronger than the Ap condition and is sharp since it fails when δ = 0.Dirección General de Investigación Científica y Técnic