Sharp Two-weight, weak-type norm inequalities for singular integral operators

Cruz-Uribe, David; Pérez, Carlos

doi:10.4310/mrl.1999.v6.n4.a4

Cited by 86 publications

(116 citation statements)

References 20 publications

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“…This extends the sharp results obtained in [11] for Calderón-Zygmund operators with smooth kernels to the setting of one-sided operators.…”

Section: 2supporting

confidence: 63%

“…This result with M F appears in [11] and here we extend it to the one-sided case. For convenience we state it in terms of M + F ; to pass to M − F one just switches the intervals of integration in the corresponding Muckenhoupt type condition.…”

Section: 3mentioning

confidence: 86%

“…The main ideas are implicit in [11], [12] and are further exploited in [10]. Below we present a proof in the one-sided case (see Theorem 3.14), which can be easily adapted to the present situation.…”

Section: Muckenhoupt Weightsmentioning

confidence: 99%

“…As observed before, this proof can be easily adapted to yield Theorem 3.5: one only needs to see that (3.3) guarantees that M F is bounded from L p (u 1−p ) into L p (v 1−p ) (see [11]). For further results and a deep treatment of extrapolation results of this kind the reader is referred to [10].…”

Section: Note That If X /mentioning

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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“…This extends the sharp results obtained in [11] for Calderón-Zygmund operators with smooth kernels to the setting of one-sided operators.…”

Section: 2supporting

confidence: 63%

Section: 3mentioning

confidence: 86%

Section: Muckenhoupt Weightsmentioning

confidence: 99%

Section: Note That If X /mentioning

confidence: 99%

See 2 more Smart Citations

Generalized Hörmander conditions and weighted endpoint estimates

2009

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“…Before that, the theorem was proved in the euclidean context by C. Pérez in [P1], and it was used in order to get sharp two weighted estimates for the HardyLittlewood maximal function. For other applications to different operators from harmonic analysis, see [P2], [P3], [P4], [CP1] and [CP2].…”

Section: Maximal Operators On Spaces Of Homogeneous Type 437mentioning

confidence: 99%