## Taiwanese Journal of Mathematics

### New Characterizations of Weighted Morrey-campanato Spaces

#### Abstract

Let $\alpha \in (0,\infty)$, $q \in [1,\infty]$, $s$ be a nonnegative integer, $\omega \in A_1(\mathbb{R}^n)$ (the class of Muckenhoupt's weights). In this paper, the authors introduce the weighted Morrey-Campanato space $L(\alpha,\,q,\,s,\,\omega;\,\mathbb{R}^n)$ and obtain its equivalence on different $q \in [1,\infty]$ and integers $s \ge \lfloor n\alpha\rfloor$ (the integer part of $n\alpha$). The authors then introduce the weighted Lipschitz space $\wedge(\alpha,\,\omega;\,\mathbb{R}^n)$ and prove that $\wedge(\alpha,\,\omega;\,\mathbb{R}^n) = L(\alpha,\,q,\,s,\,\omega;\,\mathbb{R}^n)$ when $\alpha \in (0,\infty)$, $s \ge \lfloor n \alpha \rfloor$ and $q \in [1,\infty]$. Using this, the authors further establish a new characterization of $L(\alpha,\,q,\,s,\,\omega;\,\mathbb{R}^n)$ by using the convolution $\varphi_{t_{B}} \ast f$ to replace the minimizing polynomial $P_{B}^{s}f$ on any ball $B$ of a function $f$ in its norm when $\alpha \in (0,\infty)$, $s \ge \lfloor n \alpha \rfloor$, $\omega \in A_1(\mathbb{R}^n) \cap RH_{1+1/\alpha}(\mathbb{R}^n)$ and $q \in [1,\infty]$, where $\varphi$ is an appropriate Schwartz function, $t_{B}$ denotes the radius of the ball $B$ and $\varphi_{t_{B}}(\cdot) \equiv t_{B}^{-n} \varphi(t_{B}^{-1}\cdot)$.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 1 (2011), 141-163.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406166

Digital Object Identifier
doi:10.11650/twjm/1500406166

Mathematical Reviews number (MathSciNet)
MR2780276

Zentralblatt MATH identifier
1232.42021

#### Citation

Yang, Dachun; Yang, Sibei. New Characterizations of Weighted Morrey-campanato Spaces. Taiwanese J. Math. 15 (2011), no. 1, 141--163. doi:10.11650/twjm/1500406166. https://projecteuclid.org/euclid.twjm/1500406166

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