A Fefferman–Stein inequality for the Carleson operator

Beltran, David

doi:10.4171/rmi/984

Cited by 10 publications

(22 citation statements)

References 44 publications

(131 reference statements)

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“…We would like to point out the fact that similar results for Carleson operators were obtained in [17] and later on in [3].…”

Section: Introduction and Main Resultssupporting

confidence: 71%

“…We remark that the qualitative version of Theorem 1.12 is also obtained by Beltran [3] by using two weight bump theorem. We shall give two proofs for Theorem 1.12, one using the Rubio de Francia algorithm and the other one given in the Appendix B using the two weight bump theorem.…”

Section: Introduction and Main Resultssupporting

confidence: 53%

“…We thank Francesco Di Plinio for sending us a preprint of [9] and for some useful comments calling our attention to some results from the paper by David Beltran [3]. We would also like to thank David Beltran for telling us his qualitative proof of Theorem 1.12.…”

Section: Acknowledgementsmentioning

confidence: 99%

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Weighted Norm Inequalities for Rough Singular Integral Operators

Pérez

Rivera-Rı́os

et al. 2018

J Geom Anal

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In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals T Ω with Ω ∈ L ∞ (S n−1 ) and the Bochner-Riesz multiplier at the critical index B (n−1)/2 . More precisely, we prove qualitative and quantitative versions of Coifman-Fefferman type inequalities and their vector-valued extensions, weighted A p − A ∞ strong and weak type inequalities for 1 < p < ∞, and A 1 − A ∞ type weak (1, 1) estimates. Moreover, Fefferman-Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 90's. As a corollary, we obtain the weighted A 1 − A ∞ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function Ω ∈ L q (S n−1 ), 1 < q < ∞, and provide Fefferman-Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by , results by the first author in [37], suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for A ∞ weights [13,16] and ideas contained in previous works by A. Seeger in [50] and D. Fan and S. Sato [22].

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“…We would like to point out the fact that similar results for Carleson operators were obtained in [17] and later on in [3].…”

Section: Introduction and Main Resultssupporting

confidence: 71%

Section: Introduction and Main Resultssupporting

confidence: 53%

See 1 more Smart Citation

Weighted Norm Inequalities for Rough Singular Integral Operators

Pérez

Rivera-Rı́os

et al. 2018

J Geom Anal

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“…We remark that in the case of the standard symbol class S 0 :" S 0 1,0 and the classes S 0 1,δ , with δ ă 1, the inequality (4) holds with maximal operator M 3 ; this is a consequence of a result of Pérez [31] for Calderón-Zygmund operators. We note that the number of compositions of M in (2) and (4) is unlikely to be sharp here and we do not concern ourselves with such finer points in this paper.…”

Section: Introductionmentioning

confidence: 93%

Control of pseudodifferential operators by maximal functions via weighted inequalities

Beltran

2018

Trans. Amer. Math. Soc.

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Abstract. We establish general weighted L 2 inequalities for pseudodifferential operators associated to the Hörmander symbol classes S m ρ,δ . Such inequalities allow to control these operators by fractional "non-tangential" maximal functions, and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar-Stein almost orthogonality principle and a quantitative version of the symbolic calculus.

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“…In the case of pr, 1q-sparse forms, where the condition on γ becomes γ ą 1, the first author observed in [2] that M γ may be improved to the controlling maximal function M " M tpu`1 ; here M tpu`1 denotes the ptpu`1q-fold composition of M with itself. In this case, the argument is more intrinsic and does not follow from Of course the above discussion yields Fefferman-Stein inequalities for pseudodifferential operators through the sparse bounds in Theorem 1.2.…”

Section: Fefferman-stein Inequalitiesmentioning

confidence: 99%

Sparse bounds for pseudodifferential operators

Beltran

Cladek

2020

JAMA

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We prove sparse bounds for pseudodifferential operators associated to Hörmander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.2010 Mathematics Subject Classification. Primary: 35S05, Secondary: 42B25. Key words and phrases. Pseudodifferential operators, sparse domination, weighted theory. 1 We use the notation A À B to denote that there is a constant C such that A ď CB and the notation A Àǫ B to specify the dependence of the implicit constant in a certain parameter ǫ. We omit the constant factors of π coming from our normalisation of the Fourier transform. p p,Q :" |Q|´1 ş Q |f | p dx. Given 1 ď r, s ă 8, the pr, sq sparse form Λ S,r,s is defined as Λ S,r,s pf, gq :" ÿ QPS |Q| f r,Q g s,Q , and the r-sparse operator A r,S is defined as A r,S f pxq :" ÿ QPS f r,Q χ Q pxq. Theorem 1.2. Let a P S m ρ,δ for some m ă 0 and 0 ă δ ď ρ ă 1. Then for any compactly supported bounded functions f, g on R n , there exist sparse collections S and r S of dyadic cubes such that | T a f, g | ď Cpm, ρ, r, sqΛ S,r,s 1 pf, gqand | T a f, g | ď Cpm, ρ, r, sqΛ r S,s 1 ,r pf, gq for all pairs pr, s 1 q and ps 1 , rq such that m ă´np1´ρqp1{r´1{2q, 1 ď r ď s ď 2 2 Given 1 ď r ď 8, r 1 denotes its conjugate Hölder exponent, that is 1{r`1`r 1 " 1.

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A Fefferman–Stein inequality for the Carleson operator

Cited by 10 publications

References 44 publications

Weighted Norm Inequalities for Rough Singular Integral Operators

Weighted Norm Inequalities for Rough Singular Integral Operators

Control of pseudodifferential operators by maximal functions via weighted inequalities

Sparse bounds for pseudodifferential operators

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