BMO, H^1, and Calderon-Zygmund operators for non doubling measures

Details BMO, H^1, and Calderon-Zygmund operators for non doubling measures BMO, H^1, and Calderon-Zygmund operators for non doubling measures

2000-02-18

arxiv.org/abs/math/0002152v1

Mathematics

Classical Analysis and ODEs

58 pages

Scientific paper

Given a Radon measure $\mu$ on $R^d$, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties that hold when $\mu$ is doubling remain valid for the space BMO introduced in this paper, without assuming $\mu$ doubling. For instance, Calderon-Zygmund operators which are bounded in $L^2$ are bounded from $L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp maximal function it is proved that operators bounded from $L^\infty$ into BMO and from $H^1$ into $L^1$ are also bounded on $L^p$, $1

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