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A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderón-Zygmund decomposition.

Xavier Tolsa (2001)

Publicacions Matemàtiques

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...

A remark on Fefferman-Stein's inequalities.

Y. Rakotondratsimba (1998)

Collectanea Mathematica

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.

A remark on the div-curl lemma

Pierre Gilles Lemarié-Rieusset (2012)

Studia Mathematica

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.

A sharp correction theorem

S. Kisliakov (1995)

Studia Mathematica

Under certain conditions on a function space X, it is proved that for every L ∞ -function f with ∥ f ∥ ∞ ≤ 1 one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, m e s φ ≠ 1 ≤ ɛ ∥ f ∥ 1 and ∥ φ f ∥ X ≤ c o n s t ( 1 + l o g ɛ - 1 ) . For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of L ∞ -functions on ℝ n whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.

A sharp estimate for bilinear Littlewood-Paley operator.

Lanzhe Liu (2005)

RACSAM

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.

A two weight weak inequality for potential type operators

Vachtang Michailovič Kokilashvili, Jiří Rákosník (1991)

Commentationes Mathematicae Universitatis Carolinae

We give conditions on pairs of weights which are necessary and sufficient for the operator T ( f ) = K * f to be a weak type mapping of one weighted Lorentz space in another one. The kernel K is an anisotropic radial decreasing function.

A variant sharp estimate for multilinear singular integral operators

Guoen Hu, Dachun Yang (2000)

Studia Mathematica

We establish a variant sharp estimate for multilinear singular integral operators. As applications, we obtain the weighted norm inequalities on general weights and certain L l o g + L type estimates for these multilinear operators.

A weak molecule condition for certain Triebel-Lizorkin spaces

Steve Hofmann (1992)

Studia Mathematica

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.

A₁-regularity and boundedness of Calderón-Zygmund operators

Dmitry V. Rutsky (2014)

Studia Mathematica

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...

An estimation for a family of oscillatory integrals

Magali Folch-Gabayet, James Wright (2003)

Studia Mathematica

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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