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Given a doubling measure μ on Rd, it is a classical result of harmonic analysis that Calderón-Zygmund operators which are bounded in L2(μ) are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on μ by a mild growth condition on μ. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderón-Zygmund decomposition adapted...
It is proved that, for some reverse doubling weight functions, the related operator which appears in the Fefferman Stein's inequality can be taken smaller than those operators for which such an inequality is known to be true.
We prove the div-curl lemma for a general class of function spaces, stable under the action of Calderón-Zygmund operators. The proof is based on a variant of the renormalization of the product introduced by S. Dobyinsky, and on the use of divergence-free wavelet bases.
Under certain conditions on a function space X, it is proved that for every -function f with one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, and . For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of -functions on whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.
Se establece una estimación fina para el operador bilineal de Littlewood-Paley. Como aplicación se obtienen desigualdades para la norma ponderada y estimaciones del tipo L log L para el operador bilineal.
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.
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.
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.
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...
Let K be a Calderón-Zygmund kernel and P a real polynomial defined on ℝⁿ with P(0) = 0. We prove that convolution with Kexp(i/P) is continuous on L²(ℝⁿ) with bounds depending only on K, n and the degree of P, but not on the coefficients of P.
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