Generalized Hörmander's conditions, commutators and weights

Lorente, María; Martell, José María; Torre, A. de la

doi:10.1016/j.jmaa.2008.01.003

Cited by 30 publications

(63 citation statements)

References 30 publications

(67 reference statements)

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“…However, there are examples of singular integrals like the differential transform operator from ergodic theory defined in (1.2), whose kernel K is in H r for all 1 ≤ r < ∞ but K / ∈ H ∞ . As shown in [18], K satisfies a Hörmander condition in the scale of Orlicz spaces that lies between the intersection of the classes H r for 1 ≤ r < ∞ and H ∞ . The same happens with the one-sided discrete square function considered in [36] and [20].…”

Section: Muckenhoupt Weightsmentioning

confidence: 97%

“…In [18], Coifman's type estimates were proved for commutators of singular integral operators with kernels in the following Hörmander classes that depend on the order of the commutator. Definition 3.6.…”

Section: Muckenhoupt Weightsmentioning

confidence: 99%

“…Theorem 3.11 ([20], [18]). Let T + be a one-sided singular integral operator with kernel K supported in (−∞, 0).…”

Section: 3mentioning

confidence: 99%

“…When trying to prove Coifman type estimates for T + , one finds that T + is controlled by M + s for every 1 < s < ∞. In general K / ∈ H ∞ (see [18] for the case {ν j } = {(−1) j }), thus it is not clear whether one can take s = 1, that is, whether T + behaves as a one-sided singular integral operator with smooth kernel. This motivated the new Hörmander type conditions in [20], [18]: If one shows that K belongs to some class near L ∞ then one would obtain a maximal operator near M + .…”

Section: 1mentioning

confidence: 99%

“…We also obtain weighted endpoint inequalities for the commutators of such operators with BMO functions. The corresponding Coifman estimates have been studied in [18]. One of our main examples is the differential transform operator presented above, thus we also pay attention to the one-sided operators in which case one can obtain better estimates by replacing a maximal operator by its corresponding one-sided analog.…”

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confidence: 99%

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Generalized Hörmander conditions and weighted endpoint estimates

2009

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Abstract. We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u, Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u, v) for the operators to be bounded from L p (v) to L p,∞ (u). One-sided singular integrals, like the differential transform operator, are considered as well. We also provide applications to Fourier multipliers and homogeneous singular integrals.1. Introduction. The Calderón-Zygmund decomposition is a powerful tool in harmonic analysis. Since its discovery in [6], many authors have used it to derive boundedness properties of singular integral operators. For instance, using the fact that the Hilbert and Riesz transforms are bounded on L 2 , and by means of the Calderón-Zygmund decomposition, one proves that these classical operators are of weak type (1, 1). From this starting point, in the literature one can find many boundedness results for the Hilbert and Riesz transforms: estimates on L p , one-weight and two-weight norm inequalities, etc.The Calderón-Zygmund theory generalizes these ideas to provide a general framework allowing one to deal with singular integral operators. A typical Calderón-Zygmund convolution operator T is bounded on L 2 (R n ) and has a kernel K on which various conditions are assumed. In the easiest case, K behaves as the kernel of the Hilbert or Riesz transform. That is, K decays as |x| −n and its gradient as |x| −n−1 . It was already proved in [15] that these assumptions can be relaxed if we want to show that T is of weak type (1, 1):

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Section: Muckenhoupt Weightsmentioning

confidence: 97%

Section: Muckenhoupt Weightsmentioning

confidence: 99%

“…Theorem 3.11 ([20], [18]). Let T + be a one-sided singular integral operator with kernel K supported in (−∞, 0).…”

Section: 3mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

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Generalized Hörmander conditions and weighted endpoint estimates

2009

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Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

Ibañez‐Firnkorn,

Rivera‐Ríos

2023

Mathematische Nachrichten

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In this paper, we provide mixed weak‐type inequalities generalizing previous results in an earlier work by Caldarelli and the second author and also in the spirit of earlier results by Lorente et al. One of the main novelties is that, besides obtaining estimates in the Euclidean setting, results are provided as well in spaces of the homogeneous type, being the first mixed weak‐type estimates that we are aware of in that setting.

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Sparse and weighted estimates for generalized Hörmander operators and commutators

Ibañez-Firnkorn

Rivera-Rı́os

2019

Monatsh Math

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In this paper a pointwise sparse domination for generalized Hörmander and also for iterated commutators with those operators is provided generalizing the sparse domination result in [24]. Relying upon that sparse domination a number of quantitative estimates are derived. Some of them are improvements and complementary results to those contained in a series of papers due to M. Lorente, J. M. Martell, C. Pérez, S. Riveros and A. de la Torre [30,29,28]. Also the quantitative endpoint estimates in [24] are extended to iterated commutators. Other results that are obtained in this work are some local exponential decay estimates for generalized Hörmander operators in the spirit of [34] and some negative results concerning Coifman-Fefferman estimates for a certain class of kernels satisfying particular generalized Hörmander conditions.2000 Mathematics Subject Classification. 42B20, 42B25.

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Generalized Hörmander's conditions, commutators and weights

Cited by 30 publications

References 30 publications

Generalized Hörmander conditions and weighted endpoint estimates

Generalized Hörmander conditions and weighted endpoint estimates

Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

Sparse and weighted estimates for generalized Hörmander operators and commutators

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