Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

Curbera, Guillermo P.; Garcı́a-Cuerva, José; Martell, José María; Pérez, Carlos

doi:10.1016/j.aim.2005.04.009

Cited by 96 publications

(109 citation statements)

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“…Abusing notation, as in Remark 2.1, if K ∈ H ∞ , then (2.2) holds with M A = M , where A(t) = t. This was obtained in [21] improving the corresponding result for the smaller class H * ∞ . The previous estimates are useful in applications, as T and M A have a similar behavior (see [13]). For instance, two-weight estimates can be proved in the following way:…”

Section: Muckenhoupt Weightsmentioning

confidence: 99%

“…These estimates can be seen as controlling the operator T by the HardyLittlewood maximal function M , and this allows one to show that T satisfies most of the weighted estimates that M does (see [13] for more details). When relaxing the H * ∞ condition, the operators become more singular and less smooth.…”

mentioning

confidence: 99%

“…For instance, from (1.1) one shows that T is bounded on L p (w) for 1 < p < ∞ and w ∈ A p . Also, T is of weak type (1, 1) for weights in A 1 , T is bounded on weighted rearrangement invariant function spaces, T satisfies weighted modular inequalities (see [13]), etc. All these one-weight estimates are based on the fact that (1.1) is valid for any weight in A ∞ , and the weight always varies within this class.…”

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confidence: 99%

“…Coifman's estimates are important from the points of view of weighted norm inequalities since they encode a lot of information about the singularity of the operator T (see [9] and [13]). In some sense, T behaves as the maximal operator that controls it.…”

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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Section: Muckenhoupt Weightsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Generalized Hörmander conditions and weighted endpoint estimates

2009

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“…Recently, by further developing of the extrapolation theory initiated by Rubio de Francia, Curbera, García-Cuerva, Martell and Pérez, [6,7] extend some important results in Fourier analysis, such as the study of singular integral operators, to rearrangement-invariant Banach function spaces.…”

Section: Introductionmentioning

confidence: 99%

Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces

Ho¹

2016

Ann. Acad. Sci. Fenn. Math.

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Abstract. We extend the mapping properties for the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory integral operators to rearrangementinvariant quasi-Banach function spaces. We also generalize the Fourier restriction theorem and the Sobolev embedding theorem to rearrangement-invariant quasi-Banach function spaces. We obtain the above results by introducing two families of rearrangement-invariant quasi-Banach function spaces. Furthermore, these two families of rearrangement-invariant quasi-Banach function spaces also give us some embedding and interpolation results of Triebel-Lizorkin type spaces and Hardy type spaces built on rearrangement-invariant quasi-Banach function spaces.

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Two‐weight extrapolation on function spaces and applications

Cao,

Olivo

2024

Mathematische Nachrichten

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This paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including , , and extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman–Fefferman inequalities that can be used to show some known sharp inequalities, Muckenhoupt–Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calderón–Zygmund operators. Finally, we obtain two‐weight inequalities for Littlewood–Paley operators and Fourier integral operators on weighted Banach function spaces.

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Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

Cited by 96 publications

References 41 publications

Generalized Hörmander conditions and weighted endpoint estimates

Generalized Hörmander conditions and weighted endpoint estimates

Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces

Two‐weight extrapolation on function spaces and applications

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