Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces

Bernardis, A. L.; Dalmasso, Estefanía; Pradolini, Gladis

doi:10.5186/aasfm.2014.3904

Cited by 23 publications

(27 citation statements)

References 44 publications

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“…In order to estimate σ 2 we use the inequality Proof of Theorem 2.11: We proceed by induction. We must point out that the case m = 0 was already proved in [1]. As in the proof of Theorem 2.1 we have that…”

Section: And Hölder's Inequality To Havementioning

confidence: 71%

“…We shall also need two results involving the boundedness of fractional maximal operators associated with Young functions, that can be found in [1].…”

Section: Proof Of Lemma 33mentioning

confidence: 99%

“…Hence, one of our main aims is, precisely, to give sufficient conditions on the weights in order to obtain these continuity properties. Some previous results in this direction were given in [1] where the authors study the boundedness between Lebesgue spaces with variable exponent for commutators of fractional type operators with BM O symbols, (see also [11] in the framework of Orlicz spaces).…”

Section: Introductionmentioning

confidence: 99%

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The Effect of the Smoothness of Fractional Type Operators Over Their Commutators with Lipschitz Symbols on Weighted Spaces

Dalmasso

Pradolini

Ramos

2018

FCAA

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We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including L p -L q , L p -BM O and L p -Lipschitz estimates. The kernels of such operators satisfy certain size condition and a Lipschitz type regularity, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of fractional type operators with less regular kernels satisfying a Hörmander's type inequality. As far as we know, these last results are new even in the unweighted case. Moreover, we give a characterization result involving symbols of the commutators and continuity results for extreme values of p.

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Section: And Hölder's Inequality To Havementioning

confidence: 71%

“…We shall also need two results involving the boundedness of fractional maximal operators associated with Young functions, that can be found in [1].…”

Section: Proof Of Lemma 33mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Effect of the Smoothness of Fractional Type Operators Over Their Commutators with Lipschitz Symbols on Weighted Spaces

Dalmasso

Pradolini

Ramos

2018

FCAA

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“…Proof. We adapt the argument given in [23] for the fractional maximal operator with weights (see also [7,24]). From the definition of g and the relation between p(·) and q(·) we get Thus, if we apply Hölder's inequality with 1/λ > 1 and (1/λ) ′ = 1/(1 − λ), we get…”

Section: Proof Of Theorem 16mentioning

confidence: 99%

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

et al. 2019

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We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces L p(·) w (0, ∞), assuming that the exponent function p(·) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervalsOur results extend those in [18] for the constant exponent L p spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in [42] for integral operators to the variable exponent setting.2010 Mathematics Subject Classification. Primary 42B25; Secondary 26D15, 42B35.

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“…We know that, for certain X Banach space, the kernel of the square operator satisfies the conditions S † A,X and H † A,X,k , for example X = l p and A(t) = exp By Proposition 6.9, we have that S α,X f (x) satisfies the hypothesis of Theorem 6.6. Then, Theorem 3.6 for the fractional square operator is In [1], the authors study the weights for fractional maximal operator related to Young function in the context of variable Lebesgue spaces. They characterized the weights for the boundedness of M α,A with A(t) = t r (1 + log(t)) β , r ≥ 1 and β ≥ 0.…”

Section: Fractional Integralsmentioning

confidence: 99%

Hörmander conditions for vector-valued kernels of singular integrals and their commutators

Gallo¹,

Firnkorn²

2019

Rev. Un. Mat. Argentina

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In this paper we study Coifman type estimates and weighted norm inequalities for singular integral operators T and its commutators, given by the convolution with a vector valued kernel K. We define a weaker Hörmander type condition associated with Young functions for the vector valued kernels. With this general framework we obtain as an example the result for the square operator and its commutator given in [ M. Lorente, M.S. Riveros, A. de la Torre On the Coifman type inequality for the oscillation of the one-sided averages ,(1.1) for some 0 < p < ∞ and some 0 ≤ w ∈ L 1 loc (R n ). The maximal operator M T is related to the operator T which is normally easier to deal with. In general, M T is strongly related to the kernel of T .The classical result of Coifman in [3] is, let T be a Calderón-Zygmund operator, then T is controlled by M, the Hardy-Littlewood maximal operator. In other words, for all 0 < p < ∞ and w ∈ A ∞ ,Later in [16], Rubio de Francia, Ruiz and Torrea studied operators with less regularity in the kernel. They proved that for certain operators, (1.1) holds with M T = M r f = M(|f | r ) 1/r , for some 1 ≤ r < ∞. The value of the exponent r is determined by the smoothness of the kernel, namely, the kernel satisfies an L r ′ -Hörmander condition (see the precise definition below). In [14], Martell, Pérez and Trujillo-González proved that this control is sharp in the sense that one cannot write a pointwise smaller operator M s with s < r. This yields, that for operators satisfying only the classical Hörmander condition, H 1 , the inequality (1.1) does not hold for any M r , 1 ≤ r < ∞.2010 Mathematics Subject Classification. 42B20, 42B25.

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Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces

Cited by 23 publications

References 44 publications

The Effect of the Smoothness of Fractional Type Operators Over Their Commutators with Lipschitz Symbols on Weighted Spaces

The Effect of the Smoothness of Fractional Type Operators Over Their Commutators with Lipschitz Symbols on Weighted Spaces

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

Hörmander conditions for vector-valued kernels of singular integrals and their commutators

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