Taiwanese Journal of Mathematics

New Characterizations of Weighted Morrey-campanato Spaces

Dachun Yang and Sibei Yang

Full-text: Open access


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

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

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B35: Function spaces arising in harmonic analysis 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

weighted Morrey-Campanato space weighted Lipschitz space weight


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

Export citation


  • H. Bui, M. Paluszyński and M. H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math., 119 (1996), 219-246.
  • H. Bui, M. Paluszyński and M. H. Taibleson, Characterization of Besov-Lipschitz and Triebel-Lizorkin spaces. The case $q<1$, J. Fourier Anal. Appl., 3 (1997), Special Issue, 837-846.
  • H. Bui, M. Paluszyński and M. H. Taibleson, The characterization of the Triebel-Lizorkin space for $p=\fz$, J. Fourier Anal. Appl., 6 (2000), 537-550.
  • S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa, (3), 18 (1964), 137-160.
  • D. Deng, X. T. Duong and L. Yan, A characterization of the Morrey-Campanato spaces, Math. Z., 119 (2005), 641-665.
  • J. Garc\ilz a-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. (Rozprawy Mat.), 162 (1979), 1-63.
  • J. Garc\ilz a-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985.
  • L. Grafakos, Modern Fourier Analysis, Second Edition, Graduate Texts in Math., No. 250, Springer, New York, 2008.
  • H. Greenwald, On the theory of homogeneous Lipschitz spaces and Campanato spaces, Pacific J. Math., 106 (1983), 87-93.
  • S. Janson, M. H. Taibleson and G. Weiss, Elementary characterizations of the Morrey-Campanato spaces, Lecture Notes in Math., 992 (1983), 101-114.
  • F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure. Appl., 14 (1961), 415-426.
  • S. Lu, Four Lectures on Real $H^p$ Spaces, World Scientific Publishing, Singapore, 1995.
  • C. B. Morrey, Partial regularity results for non-linear elliptic systems, J. Math. Mech., 17 (1967/1968), 649-670.
  • D. Müller and D. Yang, A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces, Forum Math., 21 (2009), 259-298.
  • J. Peetre, On the theory of ${\mathcal L}^p_\lambda$ spaces, J. Funct. Anal., 4 (1969), 71-87.
  • E. M. Stein, Harmonic Analysis, Real-variable Methods, Orthogonality, and Oscillatory Integral, Princeton Univ. Press, Princeton, New Jersey, 1993.
  • M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces. Representation theorems for Hardy spaces, pp. 67-149, Astérisque, 77, Soc. Math. France, Paris, 1980.
  • L. Tang, Some characterizations of weighted Morrey-Campanato spaces, Math. Nachr. (to appear).
  • H. Triebel, Theory of Function Spaces, Basel, Birkhäuser, 1983.
  • D. Yang and W. Yuan, A new class of function spaces connecting Triebel-Lizorkin spaces and $Q$ spaces, J. Funct. Anal., 255 (2008), 2760-2809.