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Zbl 0612.43003
Zorko, Cristina T.
Morrey space.
(English)
[J] Proc. Am. Math. Soc. 98, 586-592 (1986). ISSN 0002-9939; ISSN 1088-6826/e

For $1<p<\infty$, $\Omega$ an open and bounded subset of ${\bbfR}\sp n$ and a non-increasing and non-negative function $\phi$ defined in $(0,\rho\sb{0}]$, $\rho\sb 0=diam \Omega$, we introduce the space ${\frak M}\sp p\sb{\phi,0}(\Omega)$ of locally integrable functions satisfying $$\inf\sb{c\in {\bbfC}}\{\int\sb{B(x\sb 0,\rho)\cap \Omega}\vert f(x)- c\vert\sp p dx\}\le A\quad \vert B(x\sb 0,\rho)\vert \quad \phi\sp p(\rho)$$ for every $x\sb 0\in \Omega$, $0<\rho \le \rho\sb 0$, where $\vert B(x\sb 0,\rho)\vert$ denotes the volume of the ball centered in $x\sb 0$ and radius $\rho$. The constant $A>0$ does not depend on $B(x\sb 0,\rho).$ \par We also define the atomic space $H\sp{p,\phi}(\Omega)$ as the set of functions f(x) such that $f(x)=\sum\sb{i\in I}\lambda\sb i a\sb i(x)$ in the sense of distributions where $\lambda\sb i\in {\bbfR}$, $\sum\sb{i\in I}\vert \lambda\sb i\vert <\infty$, and $a\sb i$ are atoms satisfying a) $\sup p(a\sb i)\subset B(x\sb i,\rho\sb i)\cap \Omega$, b) $\int a\sb i(x) dx=0$, c) $\Vert a\sb i\Vert\sb p\le 1/(\vert B(x\sb i,\rho\sb i)\vert\sp{1/q} \phi (\rho\sb i))$, $1/p+1/q=1.$ \par We have: I) If $\phi$ (t) is non-increasing and $t\sp n \phi\sp q(t)$ is non-decreasing then $H\sp{p,\phi}(\Omega)$ is a Banach space. $II)\quad {\frak M}\sp p\sb{\phi,0}(\Omega)$ can be represented as the dual of $H\sp{q,\phi}(\Omega)$.
MSC 2000:
*43A15 Lp-spaces and other function spaces on groups, etc.
43A17 Analysis on ordered groups, Hp-theory
46E30 Spaces of measurable functions
26A33 Fractional derivatives and integrals (real functions)

Keywords: Morrey space; duality; atomic space

Cited in: Zbl 1046.35029

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