Two characterizations of central BMO space via the commutators of Hardy operators

2021 ◽  
Vol 33 (2) ◽  
pp. 505-529
Author(s):  
Zunwei Fu ◽  
Shanzhen Lu ◽  
Shaoguang Shi
Keyword(s):  
Kernel Function ◽  
Hardy Operator ◽  
Bmo Space ◽  
Lebesgue Spaces ◽  
Type Space ◽  
Hardy Operators ◽  
Formula Type

Abstract This article addresses two characterizations of BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} -type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} space by the boundedness of the commutators; (ii) characterization of the central BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} -closure of C c ∞ ⁢ ( ℝ n ) {C_{c}^{\infty}(\mathbb{R}^{n})} space via the compactness of the commutators. This is done by exploiting the center symmetry of Hardy operator deeply and by a more explicit decomposition of the operator and the kernel function.

scholarly journals A Characterization of Central BMO Space via the Commutator of Fractional Hardy Operator

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Lei Zhang ◽  
Shaoguang Shi
Keyword(s):  
Kernel Function ◽  
Lebesgue Space ◽  
Rough Kernel ◽  
Hardy Operator ◽  
Bmo Space ◽  
Symbol Function

This paper is devoted in characterizing the central BMO ℝn space via the commutator of the fractional Hardy operator with rough kernel. Precisely, by a more explicit decomposition of the operator and the kernel function, we will show that if the symbol function belongs to the central BMO ℝn space, then the commutator are bounded on Lebesgue space. Conversely, the boundedness of the commutator implies that the symbol function belongs to the central BMO ℝn space by exploiting the center symmetry of the Hardy operator deeply.

scholarly journals Weighted multilinear Hardy operators and commutators

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Zun Wei Fu ◽  
Shu Li Gong ◽  
Shan Zhen Lu ◽  
Wen Yuan
Keyword(s):  
Necessary Conditions ◽  
Morrey Spaces ◽  
Bmo Space ◽  
Weight Functions ◽  
Lebesgue Spaces ◽  
Sharp Bounds ◽  
Sharp Estimates ◽  
Sufficient And Necessary Conditions ◽  
Hardy Operators

AbstractIn this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann–Liouville and Weyl.

New variable martingale Hardy spaces

2021 ◽  
pp. 1-29
Author(s):  
Yong Jiao ◽  
Dan Zeng ◽  
Dejian Zhou
Keyword(s):  
Brownian Motion ◽  
Hardy Space ◽  
Hardy Spaces ◽  
Stochastic Integral ◽  
Lebesgue Spaces ◽  
Type Space ◽  
Variable Lebesgue Spaces ◽  
Martingale Inequalities

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

scholarly journals Characterization of Wave Front Sets in Fourier-Lebesgue Spaces and Its Application

2013 ◽  
Vol 56 (1) ◽  
pp. 1-17
Author(s):  
Keiichi Kato ◽  
Masaharu Kobayashi ◽  
Shingo Ito
Keyword(s):  
Wave Front ◽  
Lebesgue Spaces ◽  
Wave Front Sets

scholarly journals Weighted multilinear p-adic Hardy operators and commutators

2017 ◽  
Vol 15 (1) ◽  
pp. 1623-1634
Author(s):  
Ronghui Liu ◽  
Jiang Zhou
Keyword(s):  
Morrey Spaces ◽  
Special Structure ◽  
Euclidean Spaces ◽  
Lebesgue Spaces ◽  
Sharp Bounds ◽  
Hardy Operators

Abstract In this paper, the weighted multilinear p-adic Hardy operators are introduced, and their sharp bounds are obtained on the product of p-adic Lebesgue spaces, and the product of p-adic central Morrey spaces, the product of p-adic Morrey spaces, respectively. Moreover, we establish the boundedness of commutators of the weighted multilinear p-adic Hardy operators on the product of p-adic central Morrey spaces. However, it’s worth mentioning that these results are different from that on Euclidean spaces due to the special structure of the p-adic fields.

scholarly journals Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function inLp(·)Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Muhammad Sarwar ◽  
Ghulam Murtaza ◽  
Irshaad Ahmed
Keyword(s):  
Integral Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Fractional Integral Operator ◽  
Lebesgue Spaces ◽  
Higher Dimensional ◽  
Necessary And Sufficient ◽  
Decreasing Functions ◽  
Hardy Operators

One-weight inequalities with general weights for Riemann-Liouville transform andn-dimensional fractional integral operator in variable exponent Lebesgue spaces defined onRnare investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of nonnegative decreasing functions inLp(x)spaces.

scholarly journals Локальные гранд пространства Лебега

2021 ◽  
pp. 96-108
Author(s):  
S.G. Samko ◽  
S.M. Umarkhadzhiev
Keyword(s):  
Single Point ◽  
Type Theorem ◽  
Sobolev Type ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Hardy Operators

We introduce ``local grand'' Lebesgue spaces $L^{p),\theta}_{x_0,a}(\Omega)$, $0<p<\infty,$ $\Omega \subseteq \mathbb{R}^n$, where the process of ``grandization'' relates to a single point $x_0\in \Omega$, contrast to the case of usual known grand spaces $L^{p),\theta}(\Omega)$, where ``grandization'' relates to all the points of $\Omega$. We define the space $L^{p),\theta}_{x_0,a}(\Omega)$ by means of the weight $a(|x-x_0|)^{\varepsilon p}$ with small exponent, $a(0)=0$. Under some rather wide assumptions on the choice of the local ``grandizer'' $a(t)$, we prove some properties of these spaces including their equivalence under different choices of the grandizers $a(t)$ and show that the maximal, singular and Hardy operators preserve such a ``single-point grandization'' of Lebesgue spaces $L^p(\Omega)$, $1<p<\infty$, provided that the lower Matuszewska--Orlicz index of the function $a$ is positive. A Sobolev-type theorem is also proved in local grand spaces under the same condition on the grandizer.

scholarly journals Embeddings for monotone functions and $B_p$ weights, between discrete and continuous

2005 ◽  
Vol 97 (2) ◽  
pp. 281
Author(s):  
Josep L. Garcia-Domingo
Keyword(s):  
Hardy Operator ◽  
Monotone Functions ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Monotone Sequences ◽  
Classical Operator ◽  
Continuous Setting

We characterize the embedding for general weighted Lebesgue spaces of monotone functions by using the analog embedding for discrete monotone sequences indexed over the integers. We then use these results to obtain the boundedness of the discrete Hardy operator and to study the connections with the Hardy classical operator in the continuous setting.

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