We give conditions on pairs of weights which are necessary and sufficient for the operator to be a weak type mapping of one weighted Lorentz space in another one. The kernel is an anisotropic radial decreasing function.
Necessary conditions and sufficient conditions are derived in order that Bessel potential operator is bounded from the weighted Lebesgue spaces into when .
We establish a variant sharp estimate for multilinear singular integral operators. As applications, we obtain the weighted norm inequalities on general weights and certain type estimates for these multilinear operators.
We strengthen the Carleson-Hunt theorem by proving estimates for the -variation of the partial sum operators for Fourier series and integrals, for . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.
In this article we give a wavelet area integral characterization for weighted Hardy spaces Hp(ω), 0 < p < ∞, with ω ∈ A∞. Our wavelet characterization establishes the identification between Hp(ω) and T2p (ω), the weighted discrete tent space, for 0 < p < ∞ and ω ∈ A∞. This allows us to use all the results of tent spaces for weighted Hardy spaces. In particular, we obtain the isomorphism between Hp(ω) and the dual space of Hp'(ω), where 1< p < ∞ and 1/p +...
A weak molecule condition is given for the Triebel-Lizorkin spaces Ḟ_p^{α,q}, with 0 < α < 1 and 1 < p, q < ∞. As an easy corollary, one may deduce, by atomic-molecular methods, a Triebel-Lizorkin space "T1" Theorem of Han and Sawyer, and Han, Jawerth, Taibleson and Weiss, for Calderón-Zygmund kernels K(x,y) which are not assumed to satisfy any regularity condition in the y variable.
In this note we present a simple proof of a recent result of Mattila and Melnikov on the existence of limε→0 ∫|ζ-z|>ε (ζ - z)-1dμ(ζ) for finite Borel measures μ in the plane.
We prove a weighted vector-valued weak type (1,1) inequality for the Bochner-Riesz means of the critical order. In fact, we prove a slightly more general result.
In this paper we discuss a weighted version of Journé's covering lemma, a substitution for Whitney decomposition of an open set in R2 where squares are replaced by rectangles. We then apply this result to obtain a sharp version of the atomic decomposition of the weighted Hardy spaces Hu'p (R+2 x R+2) and a description of their duals when p is close to 1.
The Coifman-Fefferman inequality implies quite easily that a Calderón-Zygmund operator T acts boundedly in a Banach lattice X on ℝⁿ if the Hardy-Littlewood maximal operator M is bounded in both X and X'. We establish a converse result under the assumption that X has the Fatou property and X is p-convex and q-concave with some 1 < p, q < ∞: if a linear operator T is bounded in X and T is nondegenerate in a certain sense (for example, if T is a Riesz transform) then M is bounded in both X and...
A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.