scholarly journals Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

2022 ◽  
Vol 7 (4) ◽  
pp. 5971-5990
Author(s):  
Yanqi Yang ◽  
◽  
Shuangping Tao ◽  
Guanghui Lu
Keyword(s):  
Maximal Function ◽  
Lipschitz Function ◽  
Variable Exponent ◽  
Differential Operators ◽  
Sharp Maximal Function ◽  
Pointwise Estimates ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Pseudo Differential Operators

<abstract><p>In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.</p></abstract>

scholarly journals A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces

2021 ◽  
Vol 19 (1) ◽  
pp. 306-315
Author(s):  
Esra Kaya
Keyword(s):  
Differential Operator ◽  
Maximal Function ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Differential Operators ◽  
Necessary Condition ◽  
Lebesgue Spaces ◽  
Fractional Maximal Function ◽  
Variable Exponent Lebesgue Spaces ◽  
Bessel Differential Operator

Abstract In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications

2019 ◽  
Vol 31 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Yan Lin ◽  
Guozhen Lu ◽  
Shanzhen Lu
Keyword(s):  
Maximal Function ◽  
Sharp Maximal Function ◽  
Pointwise Estimates ◽  
Lebesgue Spaces ◽  
Zygmund Operator ◽  
Bmo Functions ◽  
Mean Oscillation ◽  
Size Condition ◽  
Multilinear Commutators ◽  
The Mean

Abstract In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates.

scholarly journals Multilinear strongly singular integral operators with generalized kernels and applications

2021 ◽  
Vol 6 (12) ◽  
pp. 13533-13551
Author(s):  
Shuhui Yang ◽  
◽  
Yan Lin
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Variable Exponent ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces

<abstract><p>In this paper, the authors study the boundedness properties of a class of multilinear strongly singular integral operator with generalized kernels on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. Moreover, the types $ L^{\infty}\times \dots \times L^{\infty}\rightarrow BMO $ and $ BMO\times \dots \times BMO\rightarrow BMO $ endpoint estimates are also obtained.</p></abstract>

The one-sided Ap conditions and local maximal operator

2011 ◽  
Vol 55 (1) ◽  
pp. 79-104 ◽  
Author(s):  
Ana L. Bernardis ◽  
Amiran Gogatishvili ◽  
Francisco Javier Martín-Reyes ◽  
Pedro Ortega Salvador ◽  
Luboš Pick
Keyword(s):  
Function Space ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
The One

AbstractWe introduce the one-sided local maximal operator and study its connection to the one-sided Ap conditions. We get a new characterization of the boundedness of the one-sided maximal operator on a quasi-Banach function space. We obtain applications to weighted Lebesgue spaces and variable-exponent Lebesgue spaces.

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