A note on weighted bounds for singular operators with nonsmooth kernels

Bui, The Anh; Conde-Alonso, José M.; Duong, Xuan Thinh; Hormozi, Mahdi

doi:10.4064/sm8409-9-2016

Cited by 14 publications

(17 citation statements)

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“…The missing case α " 0, β ‰ 0 in the above theorem corresponds to the classical case of fractional integrals, for which fractional sparse bounds were obtained by Cruz-Uribe and Moen in [15], and the case α " β " 0 corresponds to the classical Mikhlin multipliers; see for instance [6] for sparse bounds. To approach the α " 0 case with our methods, one needs to consider both low and high frequencies.…”

Section: Oscillatory Fourier Multipliersmentioning

confidence: 95%

“…Since a finite union of sparse collections is sparse, it suffices to prove sparse domination for ÿ jě0: j"apmod N q C´j ÿ Q dyadic: ℓpQq"2 t´jρ`jǫ`10u |Q| f r,Q g s 1 ,Q (7) for some N , and for all 0 ď a ď N´1. Choose N sufficiently large so that C N ą 4 n and N ą 100{ρ if ρ ‰ 0, so that different j in (7) correspond to different dyadic levels of cubes; this is in contrast with the situation in (6). Fix some such a.…”

Section: Proof It Suffices To Show Thatmentioning

confidence: 99%

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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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Section: Oscillatory Fourier Multipliersmentioning

confidence: 95%

Section: Proof It Suffices To Show Thatmentioning

confidence: 99%

Sparse bounds for pseudodifferential operators

Beltran

Cladek

2020

JAMA

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“…with countably many cubes Q i ∈ D 0 that satisfying conditions (2) and (3) and if the constant C T 0 is of the form c n ||T || L q 1 ×L q 2 →L q + C K + ||ω|| Dini , then the cubes also satisfy condition (1).…”

Section: Some Auxiliar Operators and A Related Lemma Let T Be A Bilimentioning

confidence: 99%

New bounds for bilinear Calderón–Zygmund operators and applications

Damián

Hormozi

2018

Rev. Mat. Iberoam.

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In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calderón-Zygmund operators with Dini-continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hytönen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple A∞ constants inspired in the Fujii-Wilson and Hrusčěv classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón-Zygmund operators and their commutators with BM O functions, square functions and multilinear Fourier multipliers.

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“…Next we will show that L r -Hörmander condition is also weaker than the regularity assumption used in [1] (which was originally introduced in [2]). Recall that the regularity assumption in [1] reads as follows:…”

Section: Some Remarksmentioning

confidence: 97%

“…It is shown in [16,Theorem 3] that T is not bounded on L p (w) for some w ∈ A p when p < n/s or p > ( n s ) ′ . This means (H2) (which is a consequence of the assumption described as above, see [1]) and therefore the L r -Hörmander condition are not sufficient for the Dini condition.…”

Section: By Triangle Inequality We Havementioning

confidence: 99%