BMO-space

space of functions of bounded mean oscillation

Functions of bounded mean oscillation were introduced by F. John and L. Nirenberg [a8], [a12], in connection with differential equations. The definition on ${\bf R} ^ { n }$ reads as follows: Suppose that $f$ is integrable over compact sets in ${\bf R} ^ { n }$, (i.e. $f \in L ^ { 1 _ {\operatorname{ loc }}} ( \mathbf{R} )$), and that $Q$ is any ball in ${\bf R} ^ { n }$, with volume denoted by $|Q|$. The mean of $f$ over $Q$ will be

\begin{equation*} f _ { Q } = \frac { 1 } { | Q | } \int _ { Q } f ( t ) d t. \end{equation*}

By definition, $f$ belongs to $\operatorname{BMO}$ if

\begin{equation*} \| f \| _ { * } = \operatorname { sup } _ { Q } \frac { 1 } { | Q | } \int _ { Q } | f ( t ) - f _ { Q } | d t < \infty, \end{equation*}

where the supremum is taken over all balls $Q$. Here, $\| f \|_*$ is called the $\operatorname{BMO}$-norm of $f$, and it becomes a norm on $\operatorname{BMO}$ after dividing out the constant functions. Bounded functions are in $\operatorname{BMO}$ and a $\operatorname{BMO}$-function is locally in $L _ { p } ( \mathbf{R} )$ for every $p < \infty$. Typical examples of $\operatorname{BMO}$-functions are of the form $\operatorname { log } | P |$ with $P$ a polynomial on ${\bf R} ^ { n }$.

The space $\operatorname{BMO}$ is very important in modern harmonic analysis. Taking $n = 1$, the Hilbert transform $H$, defined by $H f ( x ) = \operatorname { lim } _ { \epsilon \downarrow 0} \int _ { | t | > \epsilon } f ( x - t ) / t d t$, maps $L _ { \infty } ( \mathbf{R} )$ to $\operatorname{BMO}$ boundedly, i.e.

\begin{equation*} \| H f \| _ { * } \leq G \| f \| _ { \infty }. \end{equation*}

The same is true for a large class of singular integral transformations (cf. also Singular integral), including Riesz transformations [a12]. There is a version of the Riesz interpolation theorem (cf. also Riesz interpolation formula) for analytic families of operators $\{ T _ { s } \}$, $0 \leq \operatorname { Re } s \leq 1$, which besides the $L_{2}$-boundedness assumptions on $\| T _ { i t } \|$ involves the (weak) assumption $\| T _ { 1 } + i t ( f ) \| _ { * } \leq C \| f \| _ { \infty }$ instead of the usual assumption $\| T _ { 1 + i t} ( f ) \| _ { \infty } \leq C \| f \|_\infty$, cf. [a12]. However the most famous result is the Fefferman duality theorem, [a6], [a7], [a12]. It states that the dual of $H ^ { 1 }$ is $\operatorname{BMO}$. Here, $H ^ { 1 }$ denotes the real Hardy space on ${\bf R} ^ { n }$ (cf. also Hardy spaces). The result is also valid for the usual space $H ^ { 1 }$ on the disc or the upper half-plane, with an appropriate complex multiplication on $\operatorname{BMO}$, cf. [a5].

Calderón–Zygmund operators on ${\bf R} ^ { n }$ form an important class of singular integral operators. A Calderón–Zygmund operator can be defined as a linear operator $T : \mathcal{D} ( \mathbf{R} ^ { n } ) \rightarrow \mathcal{D} ^ { \prime } ( \mathbf{R} ^ { n } )$ with associated Schwarz kernel $K ( x , y )$ defined on $\Omega = \{ ( x , y ) : x , y \in \mathbf{R} ^ { n } , x \neq y \}$ with the following properties:

i) $K$ is locally integrable on $\Omega$ and satisfies $| K ( x , y ) | = O ( | x - y | ^ { - x } )$;

ii) there exist constants $C > 0$ and $0 < \gamma \leq 1$ such that for $( x , y ) \in \Omega$ and $| x ^ { \prime } - x | \leq | x - y | / 2$,

\begin{equation*} | K ( x - , y ) - K ( x , y ) | \leq C | x ^ { \prime } - x | ^ { \gamma } | x - y | ^ { - n - \gamma }. \end{equation*}

Similarly, for $( x , y ) \in \Omega$ and $| y ^ { \prime } - y | \leq | x - y | / 2$,

\begin{equation*} | K ( x , y ^ { \prime } ) - K ( x , y ) | \leq C | y ^ { \prime } - y | ^ { \gamma } | x - y | ^ { - n - \gamma }. \end{equation*}

iii) $T$ can be extended to a bounded linear operator on $L _ { 2 } ( \mathbf{R} ^ { n } )$.

This last condition is hard to verify in general. Thus, it is an important result, known as the $T ( 1 )$-theorem, that if i) and ii) hold, then iii) is equivalent to: $T$ is weakly bounded on $L_{2}$ and both $T ( 1 )$ and $T ^ { * } ( 1 )$ are in $\operatorname{BMO}$, cf. [a3], [a11], [a12]. It is known that diagonal operators with respect to an orthonormal wavelet basis are of Calderón–Zygmund type. This connection with wavelet analysis is treated in [a11].

Many of the results concerning $\operatorname{BMO}$-functions have been generalized to the setting of martingales, cf. [a9] (see also Martingale).

The duality result indicates that $\operatorname{BMO}$ plays a role in complex analysis as well. The class of holomorphic functions (cf. Analytic function) on a domain $D$ with boundary values in $\operatorname{BMO}$ is denoted by , and is called the -space, i.e., $= \operatorname{BMOA}= \operatorname{BMO} \cap H ^ { 2 }$.

Carleson's corona theorem [a5] for the disc states that for given bounded holomorphic functions $f _ { 1 } , \ldots , f _ { n }$ such that $\sum _ { i } | f _ { i } | > \delta > 0$ there exist bounded holomorphic functions $g_ 1 , \ldots , g_ { n }$ such that $\sum _ { i } f _ { i } g _ { i } = 1$. So far (1996), this result could not be extended to the unit ball in $\mathbf{C}^{m}$, $m > 1$, but it can be proved if one only requires that $g_i \in \operatorname { BMOA}$, cf. [a13].

The definition of $\operatorname{BMO}$ makes sense as soon as there are proper notions of integral and ball in a space. Thus, $\operatorname{BMO}$ can be defined in spaces of homogeneous type, cf. [a1], [a2], [a10]. In the setting of several complex variables, several types of $\operatorname{BMO}$-spaces arise on the boundary of (strictly) pseudoconvex domains, depending on whether one considers the isotropic Euclidean balls or the non-isotropic balls that are natural in connection with pseudo-convexity, cf. [a10].

References

[a1]	R.R. Coifman, G. Weiss, "Analyse harmonique non-commutative sur certains espaces homogènes" , Lecture Notes in Mathematics , 242 , Springer (1971)
[a2]	R.R. Coifman, G. Weiss, "Extensions of Hardy spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 (1977) pp. 569–643
[a3]	G. David, J.-L. Journé, "A boundedness criterion for generalized Calderón–Zygmund operators" Ann. of Math. , 120 (1985) pp. 371–397
[a4]	J. Garcia-Cuervas, J.L. Rubio de Francia, "Weighted norm inequalities and related topics" , Math. Stud. , 116 , North-Holland (1985)
[a5]	J. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a6]	C. Fefferman, "Characterizations of bounded mean oscillation" Bull. Amer. Math. Soc. , 77 (1971) pp. 587–588
[a7]	C. Fefferman, E.M. Stein, "$H ^ { p }$ spaces of several variables" Acta Math. , 129 (1974) pp. 137–193
[a8]	F. John, L. Nirenberg, "On functions of bounded mean oscillation" Comm. Pure Appl. Math. , 14 (1961) pp. 415–426
[a9]	N. Kazamaki, "Continuous exponential martingales and BMO" , Lecture Notes in Mathematics , 579 , Springer (1994)
[a10]	S.G. Krantz, "Geometric analysis and function spaces" , CBMS , 81 , Amer. Math. Soc. (1993)
[a11]	Y. Meyer, "Ondelettes et opérateurs II. Opérateurs de Calderón–Zygmund" , Actual. Math. , Hermann (1990)
[a12]	E.M. Stein, "Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals" , Math. Ser. , 43 , Princeton Univ. Press (1993)
[a13]	N.Th. Varopoulos, "BMO functions and the $\partial$ equation" Pacific J. Math. , 71 (1977) pp. 221–272