Weak and strong type estimates for the multilinear pseudo-differential operators

Cao, Mingming; Xue, Qingying; Yabuta, Kôzô

doi:10.1016/j.jfa.2019.108454

“…Beyond that, we consider other weak type estimates including the restricted weak-type (p, p) estimates and the endpoint estimate for the corresponding commutators. For more information about the progress of these estimates, see [6,14,30,31,29,32] and the reference therein.…”

Section: Introductionmentioning

confidence: 99%

Weak and strong types estimates for square functions associated with operators

Cao¹,

Si²,

Zhang³

2020

Preprint

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Let L be a linear operator in L 2 (R n ) which generates a semigroup e −tL whose kernels p t (x, y) satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function S α,L associated with an abstract operator L. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding Sawyer's conjecture by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type (p, p) for S α,L and the endpoint estimate for commutators of S α,L . Finally, all the conclusions aforementioned can be applied to a number of square functions associated to L.

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“…Indeed, it is enough to show that each operator T above verifies (5.16). This will follow from the point sparse domination obtained in [51,Theorem 1.4], [16,Theorem 1.11], [70,Proposition 3.3] and [14,Proposition 4.1] respectively.…”

Section: Extrapolation For Commutatorsmentioning

confidence: 87%

“…where c n,q is independent of Q, f and w. Indeed, using the techniques in [14,61], one can show (6.8)-(6.12). But it needs the sharp maximal function control for these operators.…”

Section: Applicationsmentioning

confidence: 94%

“…The Rubio de Francia extrapolation theorem and its variants have been proved to be extremely advantageous and the key to solving many problems in harmonic analysis. Indeed, extrapolation theorems allow us to reduce the general L p estimates for certain operators to a suitable case p = p 0 , for example, see [14] for the Coifman-Fefferman's inequality for p 0 = 1, [39] for the Calderón-Zygmund operators for p 0 = 2, and [47] for square functions for p 0 = 3. Even more, the technique of extrapolation can refine some estimates, see [20] for the Sawyer conjecture, [48,49] for the weak Muckenhoupt-Wheeden conjecture and [61] for the local decay estimates.…”

Section: Introductionmentioning

confidence: 99%

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Two-weight extrapolation on function spaces and applications

Cao¹,

Olivo²

2021

Preprint

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This paper is devoted to studying the two-weight extrapolation theory of Rubio de Francia. We start to establish endpoint extrapolation results including A1, Ap and A∞ extrapolation in the context of Banach function spaces and modular spaces. Furthermore, we present extrapolation for commutators in weighted Banach function spaces. Beyond that, we give several applications which can be easily obtained using extrapolation. First, we get local decay estimates for various operators and commutators. Second, Coifman-Fefferman inequalities are established and can be used to show some known sharp A1 inequalities. Third, Muckenhoupt-Wheeden and Sawyer 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. Contents 1. Introduction 2 2. Preliminaries 3 2.1. Muckenhoupt weights 3 2.2. Orlicz maximal operators 5 2.3. Banach function spaces 8 3. Extrapolation on Banach function spaces 14 3.1. From one-weight to two-weight 14 3.2. From two-weight to two-weight 23 4. Extrapolation on modular spaces 27 5. Extrapolation for commutators 32 6. Applications 38 6.1. Local decay estimates 39 6.2. Coifman-Fefferman inequalities 42 6.3. Muckenhoupt-Wheeden conjecture 43

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“…It is worth mentioning that we couldn't find in the literature any trace of results like (1.10) involving M or multi-linear Calderón-Zygmund operators, A R p or A R P weights, and mixed restricted weak type inequalities. Curiously, we didn't find much about Sawyer-type inequalities for Lorentz spaces apart from the endpoint results studied in [2,19,24,39,40,44,[47][48][49]54], and some endpoint estimates for multi-variable fractional operators (see [52]), multi-linear pseudo-differential operators (see [12]), and the Hardy averaging operator (see [41,43]). As we have seen before, these inequalities are fundamental to understand the behavior of the operator M ⊗ , but they appear naturally in the study of other classical operators, even in the one-variable case.…”

Section: Introductionmentioning

confidence: 99%

Sawyer-type inequalities for Lorentz spaces

Pérez

¹

,

Roure-Perdices

²

2021

Math. Ann.

5

0

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The Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{1,\infty }(uv)} \le C_{u,v} \Vert f \Vert _{L^{1}(u)}, \end{aligned}$$ Mf v L 1 , ∞ ( u v ) ≤ C u , v ‖ f ‖ L 1 ( u ) , where $$u\in A_1$$ u ∈ A 1 and $$uv\in A_{\infty }$$ u v ∈ A ∞ . We prove a novel extension of this result to the general restricted weak type case. That is, for $$p>1$$ p > 1 , $$u\in A_p^{{\mathcal {R}}}$$ u ∈ A p R , and $$uv^p \in A_\infty $$ u v p ∈ A ∞ , $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{p,\infty }(uv^p)} \le C_{u,v} \Vert f \Vert _{L^{p,1}(u)}. \end{aligned}$$ Mf v L p , ∞ ( u v p ) ≤ C u , v ‖ f ‖ L p , 1 ( u ) . From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the m-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including m-linear Calderón-Zygmund operators, avoiding the $$A_\infty $$ A ∞ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $$A_p^{{\mathcal {R}}}$$ A p R . Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $${\mathcal {M}}$$ M , denoted by $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P R , establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, $$A_p^{{\mathcal {R}}}$$ A p R and $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P R weights, and Lorentz spaces.

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Weak and strong type estimates for the multilinear pseudo-differential operators

Cited by 16 publications

References 60 publications

Weak and strong types estimates for square functions associated with operators

Weak and strong types estimates for square functions associated with operators

Two-weight extrapolation on function spaces and applications

Sawyer-type inequalities for Lorentz spaces

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