A Note on Campanato Spaces and Their Applications

Wang, Dinghuai; Zhou, Jiang; Teng, Zhidong

doi:10.1134/s0001434618030148

Cited by 11 publications

(4 citation statements)

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“…The equivalence can be found in [5, pages 14 and 38] for q = 1 and in [10] for 1 < q < ∞. Recently, we showed that the result holds for 0 < q < 1 in [16].…”

Section: Introductionsupporting

confidence: 52%

Characterizations of Bmo and Lipschitz Spaces in Terms of Weights and Their Applications

Wang¹,

Zhou²,

Teng³

2019

J. Aust. Math. Soc.

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Let $0<\unicode[STIX]{x1D6FC}<n,1\leq p<q<\infty$ with $1/p-1/q=\unicode[STIX]{x1D6FC}/n$, $\unicode[STIX]{x1D714}\in A_{p,q}$, $\unicode[STIX]{x1D708}\in A_{\infty }$ and let $f$ be a locally integrable function. In this paper, it is proved that $f$ is in bounded mean oscillation $\mathit{BMO}$ space if and only if $$\begin{eqnarray}\sup _{B}\frac{|B|^{\unicode[STIX]{x1D6FC}/n}}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty ,\end{eqnarray}$$ where $\unicode[STIX]{x1D714}^{p}(B)=\int _{B}\unicode[STIX]{x1D714}(x)^{p}\,dx$ and $f_{\unicode[STIX]{x1D708},B}=(1/\unicode[STIX]{x1D708}(B))\int _{B}f(y)\unicode[STIX]{x1D708}(y)\,dy$. We also show that $f$ belongs to Lipschitz space $Lip_{\unicode[STIX]{x1D6FC}}$ if and only if $$\begin{eqnarray}\sup _{B}\frac{1}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty .\end{eqnarray}$$ As applications, we characterize these spaces by the boundedness of commutators of some operators on weighted Lebesgue spaces.

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“…The equivalence can be found in [5, pages 14 and 38] for q = 1 and in [10] for 1 < q < ∞. Recently, we showed that the result holds for 0 < q < 1 in [16].…”

Section: Introductionsupporting

confidence: 52%

Characterizations of Bmo and Lipschitz Spaces in Terms of Weights and Their Applications

Wang¹,

Zhou²,

Teng³

2019

J. Aust. Math. Soc.

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“…This was shown independently by Campanato [3] and by Meyers [16] for 1 ≤ p < ∞, and by the first author, Zhou and Teng [25] for 0 < p < 1.…”

Section: Introductionmentioning

confidence: 57%

New function classes of Morrey-Campanato type and their applications

Wang¹,

Li-sheng²

2021

Preprint

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The aim of this paper is to introduce and investigative some new function classes of Morrey-Campanato type. Let 0 < p < ∞ and 0 ≤ λ < n + p. We say that f ∈ Lp,λ (Ω) if supwhere Ω(x 0 , ρ) = Q(x 0 , ρ) ∩ Ω and Q(x, ρ) is denote the cube of R n . Some basic properties and characterizations of these classes are presented. If 0 ≤ λ < n, the space is equivalent to related Morrey space.functions establish an integral characterization of the nonnegative Hölder continue functions. As applications, this paper gives unified criterions on the necessity of bounded commutators of maximal functions.

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“…The equivalence can be found in [3] for q = 1, [7] for 1 < q < ∞ and [15] for 0 < q < 1. Specially, C 0,q = BMO, the spaces of bounded mean oscillation.…”

Section: The Inequalitymentioning

confidence: 89%