Weighted estimates in a limited range with applications to the Bochner-Riesz operators

Carro, María Jesús Casals; Zuazo, Francisco Javier Duoandikoetxea; Lorente, María

doi:10.1512/iumj.2012.61.4723

Cited by 14 publications

(12 citation statements)

References 33 publications

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“…Concerning weighted estimates, it is well-known that linear Calderón-Zygmund operators are bounded on L p (ω) for p ∈ (1, ∞) and every weight ω belonging to the Muckenhoupt's class A p (see Definitions 1.1 and 1.2 for more details about Muckenhoupt's class A p and Reverse Hölder class RH s ). Similar properties are satisfied by the Hardy-Littlewood maximal operator and some other linear operators as Bochner-Riesz multipliers [15,4] or non-integral operators (like Riesz transforms) [1]. All these boundedness, obtained by using suitable Fefferman-Stein inequalities related to maximal sharp functions, involve weights belonging to the class W p (p 0 , q 0 ) := A p p 0 ∩ RH ( q 0 p ) ′ for some exponents p 0 < q 0 .…”

mentioning

confidence: 62%

Multi-frequency Calderón-Zygmund analysis and connexion to Bochner-Riesz multipliers

Bernicot

2014

Harmonic Analysis and Partial Differential Equations

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In this work, we describe several results exhibited during a talk at the El Escorial 2012 conference. We aim to pursue the development of a multi-frequency Calderón-Zygmund analysis introduced in [10]. We set a definition of general multi-frequency Calderón-Zygmund operators. Unweighted estimates are obtained using the corresponding multi-frequency decomposition of [10]. Involving a new kind of maximal sharp function, weighted estimates are obtained.

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confidence: 62%

Multi-frequency Calderón-Zygmund analysis and connexion to Bochner-Riesz multipliers

Bernicot

2014

Harmonic Analysis and Partial Differential Equations

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“…We remark here that the results in papers [, ] are formulated for the maximal Bochner–Riesz operator and therefore they are not expected to be optimal. Moreover, we note that the statement in (2) can be obtained through complex interpolation between the

L^{2}

weighted estimate for the critical index

δ = (n - 1) / 2

and the unweighted

L^{2}

boundedness of

B^{0}

.…”

Section: Definitions and Main Resultsmentioning

confidence: 97%

“…See Section 7 (Proposition ) for the precise equivalence. To our knowledge, the results concerning

A_{p}

weights in this range of

δ

are the following: Theorem Let

n ⩾ 2

, then: (1)Christ : If

(n - 1) / 2 (n + 1) < δ < (n - 1) / 2

, then

B^{δ}

is bounded on

L^{2} false(w false)

whenever

w^{n / (1 + 2 δ)} \in A_{1}

; (2)Carro–Duoandikoetxea–Lorente : If

0 < δ < (n - 1) / 2

, then

B^{δ}

is bounded on

L^{2} false(w false)

for every weight

w

such that

w^{(n - 1) / 2 δ} \in A_{2}

; (3)Duoandikoetxea–Moyua–Oruetxebarria‐Seijo : If

0 < δ < false(n - 1 false) / 2

, then

B^{δ}

is bounded on

L^{2} false(w false)

for every radial weight

w

such that

w^{n / (1 + 2 δ)} \in A_{2}

. …”

Section: Definitions and Main Resultsmentioning

confidence: 99%

Sparse bilinear forms for Bochner Riesz multipliers and applications

Benea

Bernicot

Luque

2017

Trans. London Math. Soc.

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“…By duality the adjoint operator is bounded on L p,w (R n ) for 1 < p ≤ γ and {w 1−p : w ∈ A p ′ /γ }, which is the same as A p ∩ RH γ ′ /(γ ′ −p) . Also operators bounded in a limited range of values of p satisfy weighted inequalities for weights in classes of such type (see [AM07] and [CDL12]). Similarly, we have sometimes the weak or strong boundedness with weights in a class of the type {w : w ν ∈ A 1 }.…”

Section: Boundedness On Morrey Spaces: General Resultsmentioning

confidence: 99%

“…The boundedness on the Morrey spaces are obtained from Corollary 4.4. Below the critical index only partial results can be obtained, but still there are some subclasses of A p weights (see [CDL12], for instance) for which the corresponding theorems in the previous section can be applied. Some weighted inequalities also hold for the maximal Bochner-Riesz operator.…”

Section: 5mentioning

confidence: 99%

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

Duoandikoetxea

Rosenthal

2017

J Geom Anal

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We prove that operators satisfying the hypotheses of the extrapolation theorem for Muckenhoupt weights are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt A1 weighted Lp spaces, which enable us to define the operators on the considered Morrey spaces by restriction. In this way, we can avoid the delicate problem of the definition of the operator, often ignored by the authors. In dealing with the extension problem through the embeddings (instead of using duality) one is neither restricted in the parameter range of the p's (in particular p = 1 is admissible and applies to weak-type inequalities) nor the operator has to be linear. Another remarkable consequence of our results is that vector-valued inequalities in Morrey spaces are automatically deduced. On the other hand, we also obtain A∞-weighted inequalities with Morrey quasinorms.2010 Mathematics Subject Classification. Primary 42B35, 46E30, 42B15, 42B20; Secondary 42B25.

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Weighted estimates in a limited range with applications to the Bochner-Riesz operators

Cited by 14 publications

References 33 publications

Multi-frequency Calderón-Zygmund analysis and connexion to Bochner-Riesz multipliers

Multi-frequency Calderón-Zygmund analysis and connexion to Bochner-Riesz multipliers

Sparse bilinear forms for Bochner Riesz multipliers and applications

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

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