An interpolation theorem between Calderón-Hardy spaces (original) (raw)

An Interpolation Theorem Between Calder´oncalder´ Calder´on-Hardy Spaces

2017

We obtain a complex interpolation theorem between weighted Calderón-Hardy spaces for weights in a Sawyer class. The technique used is based on the method obtained by J.-O. Strömberg and A. Torchinsky; however , we must overcome several technical difficulties associated with considering one-sided Calderón-Hardy spaces. Interpolation results of this type are useful in the study of weighted weak type inequalities of strongly singular integral operators.

One-sided singular integral operators on Calderón-Hardy spaces

Revista de la Unión Matemática Argentina

In [5J we have defined and studied the 1t�:;t(w) spaces for weights w belonging to the class A;; definecl by E. Sawyer, ancl where the parameter a is a positive real number. When a is a natural number, these spaces can be identifiecl with the one-sidecl Hardy space H� (w) clefinecl in [7J. This identifieation could be used to define a eontinuous extension of a one-sided regular Calderón-Zygmund operator from 1í�:;t(w) into 1t�:;t(w), when the parameter a is a natural number. In this paper, we give a direct definition of a one-sided regular Calderón-Zygmund operator on Aa n1í�:;(w), which is valid for any real number a> 0, and we prove that these operators can be extended to bouncled operators from 1í�:;t (w) into C. Segovia.

Weighted Estimates for Singular Integral Operators Satisfying Hörmander’s Conditions of Young Type

Journal of Fourier Analysis and Applications, 2005

The following open question was implicit in the literature: Are there singular integrals whose kernels satisfy the L r -Hörmander condition for any r > 1 but not the L ∞ -Hörmander condition? We prove that the one-sided discrete square function, studied in ergodic theory, is an example of a vector-valued singular integral whose kernel satisfies the L r -Hörmander condition for any r > 1 but not the L ∞ -Hörmander condition. For a Young function A we introduce the notion of L A -Hörmander. We prove that if an operator satisfies this condition, then one can dominate the L p (w) norm of the operator by the L p (w) norm of a maximal function associated to the complementary function of A, for any weight w in the A ∞ class and 0 < p < ∞. We use this result to prove that, for the one-sided discrete square function, one can dominate the L p (w) norm of the operator by the L p (w) norm of an iterate of the one-sided Hardy-Littlewood Maximal Operator, for any w in the A + ∞ class. R n K(x − y)f (y) dy , where the kernel K has bounded Fourier transform, and let Mf be the Hardy-Littlewood maximal function. A classical result of Coifman [4] states that if the kernel satisfies the Math Subject Classifications. Primary 42B20; secondary 42B25.

Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator

Israel Journal of Mathematics, 1989

In this paper we characterize the pairs of weights (u, w) for which the Hardy-Littlewood maximal operator M satisfies a weak type integral inequality of the form C f~,':,,axl>a) udx _-<~-~ fr ~lfl)wdx, with C independent of f and 2 > 0, where ~ is an N-function. Moreover, for a given weight w, a necessary and sufficient condition is found for the existence of a positive weight u such that M satisfies an integral inequality as above. Lastly, in the case u = w, we notice that the conclusion of the extrapolation theorem given by J. L. Rubio de Francia, which appeared in Am. J. Math. 106 (1984), can be strengthened to Orlicz spaces.

Weak-type operators and the strong fundamental lemma of real interpolation theory

Studia Mathematica, 2005

We prove an interpolation theorem for weak-type operators. This is closely related to interpolation between weak-type classes. Weak-type classes at the ends of interpolation scales play a similar role to that played by BMO with respect to the L p interpolation scale. We also clarify the roles of some of the parameters appearing in the definition of the weak-type classes. The interpolation theorem follows from a K-functional inequality for the operators, involving the Calderón operator. The inequality was inspired by a K-J inequality approach developed by Jawerth and Milman. We show that the use of the Calderón operator is necessary. We use a new version of the strong fundamental lemma of interpolation theory that does not require the interpolation couple to be mutually closed. 1. Introduction. Weak-type classes were defined in [9], [10], and [11]. Interpolation theorems between these classes give stronger versions of classical interpolation theorems, and calculations of the K-functionals between these classes give a systematic way of proving rearrangement-function inequalities for classical operators. These classes, when constructed at the natural ends of interpolation scales, turn out to be interesting in their own right, and imply useful generalizations of interpolation theorems. The definition of the weak-type classes, Definition 1.2 below, is quite general, and among other things yields an interpolation theorem, Theorem 1.3, that is valid only for a subinterval of the interpolation scale. In this paper we generalize the interpolation theorem. We define weak-type operators between interpolation pairs of Banach groups, and prove an interpolation theorem for these operators. Moreover we prove an inequality between the K-functionals of T a and of a, which implies the interpolation theorem and is of independent interest. We also show that the interpolation

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

Collectanea Mathematica, 2019

We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces L p(•) w (0, ∞), assuming that the exponent function p(•) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals {(0, b) : b > 0} on (0, ∞). Our results extend those in [18] for the constant exponent L p spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in [42] for integral operators to the variable exponent setting.

Weighted Hardy spaces associated to operators and boundedness of singular integrals

Arxiv preprint arXiv:1202.2063, 2012

Let (X, d, µ) be a space of homogeneous type, i.e. the measure µ satisfies doubling (volume) property with respect to the balls defined by the metric d. Let L be a non-negative self-adjoint operator on L 2 (X). Assume that the semigroup of L satisfies the Davies-Gaffney estimates. In this paper, we study the weighted Hardy spaces H p L,w (X), 0 < p ≤ 1, associated to the operator L on the space X. We establish the atomic and the molecular characterizations of elements in H p L,w (X). As applications, we obtain the boundedness on H p L,w (X) for the generalized Riesz transforms associated to L and for the spectral multipliers of L. Contents 1. Introduction 2. Preliminaries 2.1. Doubling metric spaces 2.2. Muckenhoupt weights 3. Weighted Hardy spaces associated to operators 3.1. Definition of weighted Hardy spaces 3.2. Finite propagation speed for the wave equation 3.3. Weighted tent spaces 3.4. Atomic characterization of weighted Hardy spaces H p L,w (X) 3.5. Molecular characterization of weighted Hardy spaces H p L,w (X) 4. Boundedness of singular integrals with non-smooth kernels 4.1. Generalized Riesz transforms 4.2. Boundedness of Riesz transforms associated with magnetic Schrödinger operators 4.3. Spectral multiplier theorem on H p L,w (X) References