Bounded mean oscillation

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In Harmonic analysis, a branch of mathematics, the space of functions of bounded mean oscillation (BMO), introduced by John & Nirenberg (1961), plays the same role in the theory of Hardy spaces that the space L^∞ of bounded functions plays in the theory of L^p-spaces. A general reference for functions of bounded mean oscillation is (Stein 1993, chapter IV).

A function u is said to be a BMO function if it is in $L^1_{\ell oc}(R^n)$ (=locally integrable functions) and the "mean oscillation"

$\frac{1}{|Q|}\int_{Q}|u(y)-u_Q|\,dy$

is bounded over all balls Q. The supremum is called the BMO norm of u and is denoted by ||u||_BMO. Here |Q| is the volume of Q and

$u_Q=\frac{1}{|Q|}\int_{Q} u(y)\,dy.$

is the average value of u on the ball Q.

The function ||u||_BMO becomes a norm on BMO functions after quotienting out by the constant functions (which have BMO norm 0).

Examples of BMO functions are all bounded (measurable) functions, and log(|P|) for any polynomial P that is not identically zero. BMO functions are locally L^p if 0<p<∞, but need not be locally bounded.

Fefferman (1971) showed that the BMO space is dual to H¹, the Hardy space with p=1; see (Fefferman & Stein 1972) for the proof. The pairing between f∈H¹ and g∈BMO is given by

$(f,g)=\int_{R^n}f(x)g(x)dx$

though some care is needed in defining this integral as it does not in general converge absolutely.

The space VMO of functions of vanishing mean oscillation is the closure of the continuous functions that vanish at infinity in BMO. It can also be defined as the space of functions whose "mean oscillations" on balls Q are not only bounded, but also tend to zero uniformly as the radius of the ball Q tends to 0 or infinity. The space VMO is a sort of Hardy space analogue of the space of continuous functions vanishing at infinity, and in particular the Hardy space H¹ is the dual of VMO. (Stein 1993, p. 180)

[edit] References

Fefferman, C. (1971), “Characterizations of bounded mean oscillation”, Bull. Amer. Math. Soc. 77: 587–588, MR 0280994, doi:10.1090/S0002-9904-1971-12763-5, <http://www.ams.org/bull/1971-77-04/S0002-9904-1971-12763-5/home.html>
Fefferman, C. & Stein, E.M. (1972), “H^p spaces of several variables"”, Acta Math. 129: 137–193, MR 0447953, DOI 10.1007/BF02392215
Folland, G.B. (2001), “Hardy spaces”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
John, F. & Nirenberg, L. (1961), “On functions of bounded mean oscillation”, Comm. Pure Appl. Math. 14: 785-799, MR 131498
Stein, Elias M. (1993), Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals., vol. 43., Princeton Mathematical Series, Princeton, NJ,: Princeton University Press, pp. xiv+695, MR 1232192, ISBN 0-691-03216-5

Wiegerinck, J. (2001), “BMO space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

Bounded mean oscillation

From Wikipedia, the free encyclopedia

[edit] References

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