Haar multipliers, paraproducts, and weighted inequalities

Katz, Nets Hawk; Pereyra, María

doi:10.1007/978-1-4612-2236-1_11

Cited by 14 publications

(19 citation statements)

References 13 publications

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“…Alternatively, one may establish the L 2 estimate in all weighted spaces L 2 (R n , w(x) dx) for w in the Muckenhoupt A 2 -class, with uniform dependence on the A 2 -constant, and invoke the weighted extrapolation theorem of Rubio de Francia to deduce the corresponding L p -estimates (cf. [29] for this approach). Figiel [19] has shown (based on an intermediate estimate [20], which he attributes to Bourgain) that one may replace L p (R n ) by L p (R n ; X) in (B.1) provided that X is a UMD space.…”

Section: Appendix B Carleson's Inequality and Paraproductsmentioning

confidence: 97%

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Kato's square root problem in Banach spaces

Hytönen

McIntosh

Portal

2008

Journal of Functional Analysis

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Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L p (R n ; X) of X -valued functions on R n . We characterize Kato's square root estimates √ Lu p ∇u p and the H ∞ -functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative L p space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X = C, we get a new approach to the L p theory of square roots of elliptic operators, as well as an L p version of Carleson's inequality.

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Section: Appendix B Carleson's Inequality and Paraproductsmentioning

confidence: 97%

“…We wish to mention that the proof of this inequality is significantly inspired by the work of N.H. Katz and M.C. Pereyra [29,36], although none of their specific results is explicitly needed.…”

Section: An L P Version Of Carleson's Inequalitymentioning

confidence: 97%

Kato's square root problem in Banach spaces

Hytönen

McIntosh

Portal

2008

Journal of Functional Analysis

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“…The dyadic paraproduct operator is bounded on the weighted L p (w) if and only if the weight w belongs to the Muckenhoupt class A d 2 (see [5]):…”

Section: Introductionmentioning

confidence: 99%

Linear bound for the dyadic paraproduct on weighted Lebesgue space L2(w)

Beznosova

2008

Journal of Functional Analysis

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The dyadic paraproduct is bounded in weighted Lebesgue spaces L p (w) if and only if the weight w belongs to the Muckenhoupt class A d p . However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L 2 (w) case. In this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L 2 (w) depends linearly on the A d 2 characteristic of the weight w using Bellman function techniques and extrapolate this result to the L p (w) case.

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“…To do this, we will need a modification of the stopping time from [11,13], which can be thought of as a matrix weighted adaption of the stopping time from [16,23]. Now assume that W is a matrix A p weight and that λ is large enough.…”

Section: Two Weight Characterization Of Paraproductsmentioning

confidence: 99%

Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

Isralowitz

2017

Integr. Equ. Oper. Theory

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Abstract. In this paper we characterize the two matrix weighted boundedness of commutators with any of the Riesz transforms (when both are matrix A p weights) in terms of a natural two matrix weighted BMO space. Furthermore, we identify this BMO space when p = 2 as the dual of a natural two matrix weighted H 1 space, and use our commutator result to provide a converse to Bloom's matrix A 2 theorem, which as a very special case proves Buckley's summation condition for matrix A 2 weights. Finally, we use our results to prove a matrix weighted John-Nirenberg inequality, and we also briefly discuss the challenging question of extending our results to the matrix weighted vector BMO setting.

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Haar multipliers, paraproducts, and weighted inequalities

Cited by 14 publications

References 13 publications

Kato's square root problem in Banach spaces

Kato's square root problem in Banach spaces

Linear bound for the dyadic paraproduct on weighted Lebesgue space L2(w)

Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

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