Weak and Strong Type $$A_1$$–$$A_\infty $$ Estimates for Sparsely Dominated Operators

Frey, Dorothée; Nieraeth, Zoe

doi:10.1007/s12220-018-9989-2

Cited by 20 publications

(16 citation statements)

References 37 publications

(85 reference statements)

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“…We refer the reader to [HP13] where quantitative estimates involving this condition first appeared. We also point out that estimates of this type for the limited range sparse operator in the case m = 1 have been studied in [FN19,Li17]. This condition has also been considered in the multilinear case in [DLP15].…”

Section: It Will Sometimes Also Be Useful To Redefinementioning

confidence: 91%

Quantitative estimates and extrapolation for multilinear weight classes

Nieraeth

2019

Math. Ann.

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In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including ℓ ∞ spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting.Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy-Littlewood maximal operator.As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and BMO spaces.2010 Mathematics Subject Classification. 42B20, 42B25.

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Section: It Will Sometimes Also Be Useful To Redefinementioning

confidence: 91%

Quantitative estimates and extrapolation for multilinear weight classes

Nieraeth

2019

Math. Ann.

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“…To construct the cubes in S k+1 we will use a local Calderón-Zygmund decomposition (see, e.g., [26,Lemma 4.5]) on P,ρ := {s ∈ P :…”

Section: Theorem 32 Let (S D μ) Be a Space Of Homogeneous Type Witmentioning

confidence: 99%

“…• Weak weighted L p -boundedness (including the endpoint p = p 0 ), for the sparse operators in (4.1) can be found [26,42]. • More precise bounds in terms of two-weight A p -A ∞ -characteristics for various special cases of the sparse operators in (4.1) can be found in, e.g., [23,42,45,52].…”

Section: Weighted Bounds For Sparse Operatorsmentioning

confidence: 99%

“…• More precise bounds in terms of two-weight A p -A ∞ -characteristics for various special cases of the sparse operators in (4.1) can be found in, e.g., [23,42,45,52]. • Weighted bounds for the fractional sparse operators in Theorem 3.4 can be found in [23] • Weighted bounds for the sparse forms in Theorem 3.5 can be found in [6,26].…”

Section: Weighted Bounds For Sparse Operatorsmentioning

confidence: 99%

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On Pointwise $$\ell ^r$$-Sparse Domination in a Space of Homogeneous Type

Lorist

2020

J Geom Anal

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We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual $$\ell ^1$$ ℓ 1 -sum in the sparse operator is replaced by an $$\ell ^r$$ ℓ r -sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the $$A_2$$ A 2 -theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

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“…One of the main consequences of sparse domination is that it provides weighted inequalities with an explicit dependence on the constant weight. For a summary of this and other applications, we refer the reader to [2,Section 4] and references therein, see also [22]. Quantitative weighted estimates may be deduced for operators dominated by bilinear sparse forms as a consequence of the following lemma.…”

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confidence: 99%

Quantitative Weighted Estimates for Rubio de Francia’s Littlewood–Paley Square Function

2019

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We consider the Rubio de Francia's Littlewood-Paley square function associated with an arbitrary family of intervals in R with finite overlapping. Quantitative weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight [w] A p/2 turns out to be sharp for 3 ≤ p < ∞, whereas the sharpness in the range 2 < p < 3 remains as an open question. The results arise as a consequence of a sparse domination shown for these operators, obtained by suitably adapting the ideas coming from [11] and [16]. Finally, some conjectures are stated in relation with the problem under study.

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Weak and Strong Type $$A_1$$–$$A_\infty $$ Estimates for Sparsely Dominated Operators

Cited by 20 publications

References 37 publications

Quantitative estimates and extrapolation for multilinear weight classes

Quantitative estimates and extrapolation for multilinear weight classes

On Pointwise $$\ell ^r$$-Sparse Domination in a Space of Homogeneous Type

Quantitative Weighted Estimates for Rubio de Francia’s Littlewood–Paley Square Function

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