BMO room

The BMO room is an object from harmonic analysis , a branch of mathematics . The abbreviation BMO stands for " bounded mean oscillation ". The function room BMO was introduced in 1961 by Fritz John and Louis Nirenberg . This space is a dual space to the real Hardy space ( Charles Fefferman , Elias Stein 1972). ${\displaystyle H^{1}(\mathbb {R} ^{n})}$

Definitions

Sharp function

Let be a locally integrable function , then is defined by ${\displaystyle f\in L_{loc}^{1}(\mathbb {R} ^{n})}$ ${\displaystyle f^{\sharp }}$

${\displaystyle f^{\sharp }(x)=\sup _{\{B|x\in B\}}{\frac {1}{\mu (B)}}\int _{B}|f(y)-f_{B}|\mathrm {d} \mu (y),}$

whereby the supremum is formed over all balls that contain. With becomes the mean value integral ${\displaystyle B}$ ${\displaystyle x}$ ${\displaystyle f_{B}}$

${\displaystyle f_{B}={\frac {1}{\mu (B)}}\int _{B}f(z)\mathrm {d} \mu (z)}$

designated.

BMO room

A function that can be integrated locally is called the BMO function, if it is restricted. In order to obtain a norm on this function space, all constant functions are identified with one another and set ${\displaystyle f}$ ${\displaystyle f^{\sharp }}$

${\displaystyle \|f\|_{BMO}=\|f^{\sharp }\|_{L^{\infty }(\mathbb {R} ^{n})}.}$

If the constant functions were not identified with one another, it would only be a semi-norm , i.e. not definitive. With this standard, the BMO room becomes a Banach room . Examples of BMO functions are all restricted, measurable functions and for a polynomial P which is not identical to zero. ${\displaystyle \|.\|_{BMO}}$ ${\displaystyle \log(|P|)}$

Duality of H ¹ and BMO

Charles Fefferman showed in 1971 that the BMO space is a dual space of , the real Hardy space with p = 1. The pairing between and is given by ${\displaystyle H^{1}}$ ${\displaystyle f\in H^{1}}$ ${\displaystyle g\in BMO}$

${\displaystyle T_{g}(f)=(f,g)=\int _{\mathbf {R} ^{n}}f(x)g(x)\,\mathrm {d} x.}$

Then the mapping is a Banach space isomorphism (not isometric ), in this sense it is dual space of . ${\displaystyle BMO\rightarrow (H^{1})',g\mapsto T_{g}}$ ${\displaystyle BMO}$ ${\displaystyle H^{1}}$

However, the above integral expression must be carefully defined, since this integral generally does not converge absolutely. However, there is a dense subspace on which the integral converges absolutely. With the help of Hahn-Banach's theorem , one can then continue the functional entirely . The room for the H ¹ functions with a compact carrier and with can be selected as the room . This is exactly the subspace that has a finite atomic decomposition . An important consequence that results from the proof of duality is the following inequality, which applies to and : ${\displaystyle f\in H^{1}}$ ${\displaystyle H_{a}^{1}}$ ${\displaystyle H^{1}}$ ${\displaystyle H_{a}^{1}}$ ${\displaystyle \textstyle \int _{\mathbb {R^{n}} }f(x)\mathrm {d} x=0}$ ${\displaystyle f\in H^{1}}$ ${\displaystyle g\in BMO}$

${\displaystyle \int _{\mathbf {R} ^{n}}f(x)g(x)\,\mathrm {d} x\leq c\int _{\mathbf {R} ^{n}}M_{\Phi }(f)(x)\mathrm {d} x\int _{\mathbb {R} ^{n}}g^{\sharp }(x)\,\mathrm {d} x=c\|f\|_{H^{1}(\mathbb {R} ^{n})}\|g\|_{BMO}}$.

Here is the non-tangential maximum function . ${\displaystyle M_{\Phi }(\cdot )}$

literature

• Elias M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , Princeton University Press 1993, ISBN 0-691-03216-5

Individual evidence

1. ↑ Announced 1971 by Fefferman Characterization of bounded mean oscillation , Bulletin AMS, Volume 77, 1971, p. 587/8 ( Memento of March 4, 2016 in the Internet Archive ). The article by Fefferman, Stein appeared in Acta Mathematica in 1972.