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 $\varphi_{t_{B}}(\cdot) \equiv t_{B}^{-n} \varphi(t_{B}^{-1}\cdot)$.