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Bounded mean oscillation

In mathematical analysis, the space of functions of bounded mean oscillation (BMO) consists of locally integrable functions f on \mathbb{R}^n such that the supremum, taken over all balls B \subset \mathbb{R}^n, of the mean absolute deviation \frac{1}{|B|} \int_B |f(x) - f_B| \, dx—where f_B denotes the average value of f over B and |B| is the Lebesgue measure of B—is finite; this supremum defines the BMO semi-norm \|f\|_{\mathrm{BMO}}.^[1] The space BMO, which is scale-invariant and satisfies the triangle inequality, captures functions with controlled local oscillations but allows unbounded growth, distinguishing it from L^\infty spaces.^[1] Introduced by Fritz John and Louis Nirenberg in 1961, BMO arose in the study of quasiconformal mappings and elliptic partial differential equations, where the authors established the John-Nirenberg inequality: for a BMO function on a cube Q, the measure of the set where |f - f_Q| > \lambda decays exponentially in \lambda, providing sharp control on how much such functions can deviate from their local averages.^[2] This inequality implies that bounded functions belong to BMO with \|f\|_{\mathrm{BMO}} \leq 2 \|f\|_{L^\infty}, and BMO contains non-trivial examples like \log |x| on \mathbb{R}^n \setminus \{0\}, which exhibit logarithmic growth but bounded mean deviations.^[1] BMO gained prominence in harmonic analysis through Charles Fefferman's 1971 characterization, proving that BMO is the dual space of the real Hardy space H^1(\mathbb{R}^n) under the pairing \langle f, g \rangle = \int_{\mathbb{R}^n} f g \, dx, with equivalent norms; this duality links BMO to atomic decompositions of H^1 and underpins many estimates in singular integral theory.^[3] Consequently, Calderón-Zygmund singular integral operators extend continuously from L^\infty to BMO, enabling the analysis of maximal functions and Riesz transforms on unbounded functions.^[1] Extensions of BMO appear in diverse settings, including on manifolds, trees, and product spaces, with applications to PDEs, probability, and geometric measure theory.^[4]^[5]

Definition and fundamentals

Historical overview

The concept of bounded mean oscillation emerged in the context of partial differential equations during the early 1960s. In 1961, Fritz John and Louis Nirenberg introduced the idea while investigating the regularity of solutions to elliptic PDEs, where they quantified the oscillation of functions relative to their means over domains to establish bounds on solution growth.^[6] Their seminal paper formalized this notion, proving key inequalities that controlled exponential growth in oscillations, laying the groundwork for the space now known as BMO.^[6] Preceding this, Sergio Campanato's work in the early 1960s provided influential precursors through his development of Campanato spaces, introduced in 1963 as a family of functional spaces generalizing Hölder continuity and addressing integrability conditions in elliptic problems. These spaces, defined via seminorms measuring deviations from means over balls, captured similar oscillatory behaviors and later revealed BMO as a specific instance (corresponding to certain parameters in the Campanato scale). Campanato's contributions, building on Morrey's earlier spaces from 1938, emphasized applications to higher-order regularity in PDEs. The 1970s marked a pivotal advancement when Charles Fefferman established BMO's role in harmonic analysis. In 1971, Fefferman proved that BMO is the dual space of the real Hardy space H¹, providing a functional analytic characterization that connected mean oscillation to singular integrals and maximal functions. This duality theorem, detailed further in joint work with Elias Stein in 1972, transformed BMO into a cornerstone of modern analysis, influencing areas like Littlewood-Paley theory and operator bounds.

Core definition

In mathematical analysis, particularly in the study of function spaces, the concept of bounded mean oscillation (BMO) arises in the context of locally integrable functions on Euclidean space. A function f: \mathbb{R}^n \to \mathbb{R} (or \mathbb{C}) is locally integrable, denoted f \in L^1_{\mathrm{loc}}(\mathbb{R}^n), if it is Lebesgue integrable over every compact subset of \mathbb{R}^n; the Lebesgue integral extends the Riemann integral to a broader class of functions using measure theory, allowing integration over sets of finite measure. This local integrability ensures that averages of f over bounded regions are well-defined without requiring global integrability.^[7] The space BMO(\mathbb{R}^n), introduced by John and Nirenberg, consists of all locally integrable functions f for which the mean oscillation over balls (or equivalently, cubes) is bounded. Specifically, f \in \mathrm{BMO}(\mathbb{R}^n) if

\|f\|_{\mathrm{BMO}} = \sup_B \frac{1}{|B|} \int_B |f(x) - f_B| \, dx < \infty,

where the supremum is taken over all balls B \subset \mathbb{R}^n, |B| denotes the Lebesgue measure (volume) of B, and f_B = \frac{1}{|B|} \int_B f(x) \, dx is the average value of f over B. This norm measures the supremum of the average deviation of f from its local mean, capturing how much f oscillates around its averages on scales of varying sizes. Equivalently, the supremum can be taken over cubes Q \subset \mathbb{R}^n instead of balls, yielding norms that differ by at most a dimension-dependent constant, thus defining the same space.^[8]^[7]^[9] The term "bounded mean oscillation" reflects this control: the mean absolute deviation from local averages remains uniformly bounded across all locations and scales, distinguishing BMO functions from those in L^\infty (which have bounded pointwise values) while allowing controlled growth or singularities. Constants are identified with the zero function in the BMO quotient space, as adding a constant does not change the oscillation. This formulation provides a natural extension beyond L^p spaces for p < \infty, motivated originally by solutions to elliptic partial differential equations.^[8]^[7]

Notation and equivalents

In the context of \mathbb{R}^n, the space BMO(\mathbb{R}^n) consists of locally integrable functions f for which the semi-norm

\|f\|_{\mathrm{BMO}} = \sup_{Q} \frac{1}{|Q|} \int_Q |f(x) - f_Q| \, dx < \infty

is finite, where the supremum is over all cubes Q \subset \mathbb{R}^n of positive side length, |Q| denotes the Lebesgue measure of Q, and f_Q = \frac{1}{|Q|} \int_Q f(y) \, dy is the mean value of f over Q. This semi-norm, introduced by John and Nirenberg, ignores additive constants, as \|f + c\|_{\mathrm{BMO}} = \|f\|_{\mathrm{BMO}} for any constant c \in \mathbb{R}. The notation emphasizes cubes for their dyadic structure, facilitating estimates in harmonic analysis, though the side length condition ensures the supremum captures oscillation at all scales.^[10] An equivalent characterization replaces the mean deviation around f_Q with the minimal L¹-deviation over constants: f \in \mathrm{BMO}(\mathbb{R}^n) if and only if

\sup_Q \inf_{c \in \mathbb{R}} \frac{1}{|Q|} \int_Q |f(x) - c| \, dx < \infty,

where the constants of equivalence depend only on n; this follows from the fact that the mean f_Q nearly minimizes the L¹-deviation, up to a dimension-dependent factor.^[11] Since constants lie in L^\infty(\mathbb{R}^n), this formulation aligns with deviations around bounded approximants, linking historically to Campanato spaces where such infima define Lipschitz-like regularity.^[11] The cube-based definition is equivalent to one using balls B \subset \mathbb{R}^n:

\sup_B \frac{1}{|B|} \int_B |f(x) - f_B| \, dx < \infty,

with f_B the mean over B and |B| its measure; the norms differ by a constant depending solely on n, due to the comparable geometry and measure overlap between cubes and balls in \mathbb{R}^n.^[9] A related grand mean oscillation characterization uses centered balls:

\sup_{x \in \mathbb{R}^n} \sup_{r > 0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y) - f_{B(x,r)}| \, dy < \infty,

which is comparable to \|f\|_{\mathrm{BMO}} up to universal constants, as the supremum over all balls is controlled by the centered case via covering arguments.^[10] These reformulations preserve the semi-norm structure and are essential for applications involving maximal operators or non-centered domains.^[10]

Core properties

Local p-integrability

A key integrability property of functions in BMO(ℝⁿ) is that they belong to Lᵖ_loc(ℝⁿ) for every 1 ≤ p < ∞. That is, for any f ∈ BMO(ℝⁿ) and any compact set K ⊂ ℝⁿ, the integral ∫K |f(x)|ᵖ dx is finite, and it admits an upper bound of the form C ||f||{BMO}ᵖ |K|, where C depends only on p and n. This local control distinguishes BMO from spaces like L^∞, providing uniform bounds on oscillations over cubes while allowing functions to grow at infinity.^[8] The case p = 1 follows directly from the definition of BMO, as the space consists of locally integrable functions with bounded mean oscillation over cubes. For 1 < p < ∞, the result follows from the John-Nirenberg inequality, which implies a reverse Hölder inequality bounding the Lᵖ deviations uniformly. To sketch the proof for p=1, cover the compact K with finitely many cubes {Qⱼ} of comparable size whose union contains K and whose total measure is controlled by |K|. On each Qⱼ, let aⱼ denote the average of f over Qⱼ. By the triangle inequality, |f(x)| ≤ |f(x) - aⱼ| + |aⱼ| for x ∈ Qⱼ. Integrating over Qⱼ gives ∫{Qⱼ} |f| ≤ ∫{Qⱼ} |f - aⱼ| + |aⱼ| |Qⱼ|. The BMO condition ensures ∫{Qⱼ} |f - aⱼ| ≤ ||f||{BMO} |Qⱼ|, and the second term is finite by local L¹ integrability. Summing over the finite collection yields the result for K. For p>1, the full Lᵖ bound requires the exponential decay from John-Nirenberg.^[8] This integrability highlights how BMO properly contains L^∞(ℝⁿ): any bounded function f satisfies |f - aⱼ| ≤ 2 ||f||{L^∞} on Qⱼ, so its mean oscillation is at most 2 ||f||{L^∞}, placing it in BMO with comparable norm. However, BMO extends beyond bounded functions; for instance, log |x| belongs to BMO(ℝⁿ) despite being unbounded. Constant functions exemplify the inclusion, as their mean oscillation vanishes, yielding BMO norm zero. Notably, this local control does not imply global Lᵖ membership for p < ∞; the example log |x| illustrates this, as ∫_{ℝⁿ} |log |x||ᵖ dx diverges for all such p due to slow growth at infinity.^[8]^[12]

Banach space structure

The space of functions of bounded mean oscillation (BMO) consists of all locally integrable functions f on \mathbb{R}^n such that the semi-norm

\|f\|_* = \sup_Q \frac{1}{|Q|} \int_Q |f(x) - f_Q| \, dx < \infty,

where the supremum is taken over all cubes Q \subset \mathbb{R}^n (up to negligible sets) and f_Q denotes the average of f over Q, is finite. BMO is closed under pointwise addition and scalar multiplication, forming a vector space over \mathbb{R}. The semi-norm \|\cdot\|_* satisfies the subadditivity property \|f + g\|_* \leq \|f\|_* + \|g\|_* and homogeneity \|\lambda f\|_* = |\lambda| \|f\|_* for \lambda \in \mathbb{R}, endowing BMO with the structure of a semi-normed vector space. The semi-norm \|\cdot\|_* vanishes on constant functions, so it does not separate points and BMO is initially only quasi-Banach. To resolve this and obtain a true norm, one standard approach is to consider the quotient space \mathrm{BMO}/\mathbb{R}, where \mathbb{R} identifies constant functions, equipped with the quotient semi-norm, which becomes a norm on the quotient. An equivalent construction defines a norm on the full space BMO by augmenting the semi-norm with a term fixing the constant, such as

\|f\|_{\mathrm{BMO}} = \left| \frac{1}{|Q_0|} \int_{Q_0} f(x) \, dx \right| + \|f\|_*,

where Q_0 is a fixed reference cube (e.g., the unit cube centered at the origin); this norm is independent of the choice of Q_0 up to equivalence. With this norm, BMO is a Banach space. The completeness of BMO under \|\cdot\|_{\mathrm{BMO}} (or equivalently in the quotient) is established by verifying that Cauchy sequences converge in the norm. For a Cauchy sequence \{f_k\} in BMO, the sequence converges locally in L^1 to a limit f \in L^1_{\mathrm{loc}}(\mathbb{R}^n) on each cube, since BMO embeds into L^1_{\mathrm{loc}}. To show f \in BMO, the oscillations \|f_k - f\|_* \to 0 are controlled uniformly over all cubes using the Vitali covering lemma to select subcollections of cubes where the f_k behave well, ensuring the mean oscillations of f remain bounded by the Cauchy property. Alternatively, bounds via the Hardy-Littlewood maximal function provide the necessary control on the deviations. This direct proof avoids duality arguments and confirms the Banach space structure essential for applications in functional analysis and harmonic analysis.

Comparability of cube averages

One key geometric property of functions in the space of bounded mean oscillation (BMO) on \mathbb{R}^n is the controlled difference in averages over adjacent cubes. Specifically, if Q_1 and Q_2 are cubes of equal side length sharing an (n-1)-dimensional face, then the difference of their averages satisfies |f_{Q_1} - f_{Q_2}| \leq 4 \|f\|_{\mathrm{BMO}}. This bound holds with a constant independent of the dimension n and the location or size of the cubes, reflecting the scale-invariance inherent in the BMO norm. To establish this, consider a cube Q_0 formed by the union of Q_1 and Q_2, which is bisected by their shared face. The mean oscillation of f over each half of Q_0 (corresponding to Q_1 and Q_2) is at most \|f\|_{\mathrm{BMO}}, implying |f_{Q_j} - f_{Q_0}| \leq 2 \|f\|_{\mathrm{BMO}} for j=1,2. The triangle inequality then yields the desired estimate. This approach relies on the core definition of BMO via cube averages and provides a foundation for controlling oscillations in geometric configurations crucial for multidimensional proofs. The property extends naturally to dyadic grids in \mathbb{R}^n, where averages over parent and child cubes differ by at most a constant multiple of the BMO norm. For a dyadic cube Q and one of its child subcubes Q' \subset Q (with side length half that of Q), |f_{Q'} - f_Q| \leq C_n \|f\|_{\mathrm{BMO}} with C_n depending on n (e.g., C_n = 2^n). Here, the proof decomposes Q into its $2^n equal child cubes and uses the bounded mean oscillation over Q to control deviations, leveraging the fact that f_Q is a convex combination of the child averages. This uniformity in scaling ensures the estimate applies across all levels of the dyadic decomposition, though the constant grows with dimension. These comparability results underscore the local uniformity of BMO functions over cube structures, facilitating extensions to more complex domains and operators in harmonic analysis.

John-Nirenberg inequality

The John-Nirenberg inequality provides an exponential bound on the measure of the set where a BMO function deviates significantly from its local average, highlighting the space's stronger control over tails compared to mere L^p integrability. Specifically, for f \in \text{BMO}(\mathbb{R}^n), there exist constants C_1, C_2 > 0 depending on n such that for any cube Q \subset \mathbb{R}^n and \lambda > 0,

\frac{|\{ x \in Q : |f(x) - f_Q| > \lambda \}|}{|Q|} \leq C_1 \exp\left( -\frac{C_2 \lambda}{\|f\|_{\text{BMO}}} \right),

where f_Q = \frac{1}{|Q|} \int_Q f(x) \, dx. This estimate was established by Fritz John and Louis Nirenberg in their foundational 1961 paper introducing BMO spaces, where it served as a key tool for deriving a priori estimates in elliptic partial differential equations.^[2] The proof proceeds by normalizing \|f\|_{\text{BMO}} = 1 and applying a Calderón-Zygmund-type decomposition on Q, which splits f - f_Q into a bounded part and a sum over disjoint maximal subcubes Q_j where the average exceeds a threshold, with \sum |Q_j| \leq \frac{1}{2} |Q|. Iterating this process on each Q_j generates a dyadic tree of subcubes, reducing the measure of the "bad" set exponentially at each level (with the factor depending on the subdivision into $2^n subcubes), which integrates to the desired exponential decay. This self-improving mechanism exploits the comparability of averages over adjacent cubes.^[13] A direct consequence is the reverse Hölder inequality for local averages: for $1 < p < \infty, there exists C_p > 0 (depending on n and p) such that

\left( \frac{1}{|Q|} \int_Q |f(x) - f_Q|^p \, dx \right)^{1/p} \leq C_p \|f\|_{\text{BMO}}

for every cube Q, implying that BMO functions are locally integrable to any finite power with bounds controlled by the BMO norm. This follows from integrating the distribution function estimate via the formula for L^p norms in terms of tails.^[14]

Duality and key relations

Duality with H¹ space

A pivotal result in real-variable harmonic analysis establishes that the space of functions of bounded mean oscillation, BMO(ℝⁿ), is the dual space of the real Hardy space H¹(ℝⁿ). Specifically, Fefferman proved that every continuous linear functional on H¹(ℝⁿ) can be represented uniquely as integration against a function in BMO(ℝⁿ), with the dual norm given by

\left| \int_{\mathbb{R}^n} f(x) h(x) \, dx \right| \leq \|f\|_{\mathrm{BMO}} \|h\|_{H^1}

for all f \in \mathrm{BMO}(\mathbb{R}^n) and h \in [H^1](/page/Hardy_space)(\mathbb{R}^n). The real Hardy space H^1(\mathbb{R}^n) consists of tempered distributions whose grand maximal function belongs to L^1(\mathbb{R}^n), or equivalently, those admitting an atomic decomposition into H^1-atoms—functions supported on balls with vanishing mean, bounded by the reciprocal of the ball's measure, and integrable to L^1 with finite norm. An alternative characterization involves the Riesz transforms: a function f \in L^1(\mathbb{R}^n) belongs to H^1(\mathbb{R}^n) if and only if f and its Riesz transforms are in L^1(\mathbb{R}^n), with the H^1-norm comparable to the sum of their L^1-norms. The proof of the duality proceeds in two main steps. First, one verifies that every f \in \mathrm{BMO}(\mathbb{R}^n) induces a bounded linear functional on H^1(\mathbb{R}^n) via integration, leveraging the John-Nirenberg inequality to control the pairing on H^1-atoms and the density of finite atomic combinations in H^1(\mathbb{R}^n). The converse follows from the closed graph theorem, showing that every bounded functional on H^1(\mathbb{R}^n) arises from some f \in \mathrm{BMO}(\mathbb{R}^n). This duality holds specifically for the real-variable Hardy space H^1(\mathbb{R}^n); an analogous identification fails for the complex Hardy space, underscoring the importance of the real structure in applications to singular integral operators and maximal functions in harmonic analysis.

Distance to L∞ via John-Nirenberg

The distance from a function f \in \mathrm{BMO}(\mathbb{R}^n) to the subspace L^\infty(\mathbb{R}^n) is defined as

\dist(f, L^\infty) = \inf_{g \in L^\infty} \|f - g\|_{\mathrm{BMO}}.

This quantity measures the minimal BMO norm required to approximate f by a bounded function and provides a natural metric for assessing how "far" f is from being essentially bounded.^[15] Garnett and Jones utilized the John-Nirenberg inequality to derive a precise characterization of this distance. Specifically, they proved that \dist(f, L^\infty) is comparable to the infimum of constants \varepsilon > 0 for which the exponential tail estimate in the John-Nirenberg inequality holds for f, namely,

\frac{|\{ x \in Q : |f(x) - f_Q| > \lambda \}|}{|Q|} \leq C \exp\left( -\frac{\lambda}{\varepsilon \|f\|_{\mathrm{BMO}}} \right)

for all cubes Q \subset \mathbb{R}^n and \lambda > 0, where C is a universal constant and f_Q denotes the average of f over Q. There exist absolute constants c_1, c_2 > 0 such that

c_1 \varepsilon(f) \leq \dist(f, L^\infty) \leq c_2 \varepsilon(f),

where \varepsilon(f) is this infimum. This equivalence arises from the exponential integrability provided by the John-Nirenberg inequality, which controls the distribution of deviations and yields bounds on the necessary adjustment to make f bounded.^[15] An alternative formulation equates \dist(f, L^\infty) (up to logarithmic equivalence) to

\sup_{Q} \inf_{c \in \mathbb{R}} \frac{1}{|Q|} \int_Q |f(x) - c| \, dx,

where the supremum is over all cubes Q \subset \mathbb{R}^n. The exponential tails from the John-Nirenberg inequality imply that the mean absolute deviation over Q is bounded by a multiple of \varepsilon(f), specifically \frac{1}{|Q|} \int_Q |f - f_Q| \lesssim \varepsilon(f), and optimizing over constants c (such as medians or truncated means) refines this to capture the distance precisely. This connection highlights how local oscillations prevent approximation by bounded functions unless they decay sufficiently fast.^[15] This characterization implies that L^\infty consists precisely of those functions in BMO for which \dist(f, L^\infty) = 0, confirming that bounded functions form a closed subspace of BMO under the BMO norm. Moreover, functions like the logarithm, such as f(x) = \log |x| on \mathbb{R}^n \setminus \{0\}, belong to BMO but exhibit positive distance to L^\infty, as their unbounded growth leads to persistently large local oscillations that cannot be compensated by any bounded perturbation in the BMO sense. Log-Lipschitz functions, which satisfy |\nabla f(x)| \lesssim 1/|x|, similarly have positive distance, illustrating the strict inclusion L^\infty \subsetneq \mathrm{BMO}.^[15]

Connection to Hilbert transform

The space of functions of bounded mean oscillation (BMO) provides a precise characterization of the boundedness properties of the Hilbert transform and related singular integral operators on L²(ℝ). A foundational result is that the Hilbert transform H, defined by

Hf(x) = \frac{1}{\pi} \mathrm{p.v.} \int_{\mathbb{R}} \frac{f(y)}{x - y} \, dy,

maps L^∞(ℝ) boundedly into BMO(ℝ), with operator norm comparable to its L² boundedness constant. This mapping property, established by Stein, highlights BMO as the natural "endpoint" space for the image of bounded functions under H, beyond which the operator fails to preserve L^p norms for p < ∞ except p = 2. The Fefferman-Stein decomposition theorem further elucidates this connection, stating that BMO(ℝ) consists precisely of functions expressible as f = g + Hf_1, where g, f_1 ∈ L^∞(ℝ). Announced in 1971, this characterization implies that H extends boundedly from BMO(ℝ) to itself, as applying H to both sides yields Hf = Hg - f_1, with Hg ∈ BMO(ℝ) and f_1 ∈ L^∞(ℝ) ⊂ BMO(ℝ).^[16] The decomposition arises from the duality between BMO and the real Hardy space H¹(ℝ), leveraging the fact that H preserves the atomic structure of H¹ under its adjoint action. This framework generalizes to Calderón-Zygmund operators T, which are singular integrals with smooth, size-vanishing kernels satisfying standard estimates. Such an operator T is bounded on L²(ℝ^n) if and only if it maps L^∞(ℝ^n) boundedly into BMO(ℝ^n). The forward implication follows from kernel bounds and the John-Nirenberg inequality controlling oscillations, while the converse relies on the closed graph theorem and density of smooth compactly supported functions in L².^[11] This equivalence resolves the boundedness of T on L^p(ℝ^n) for 1 < p < ∞ via real interpolation between L¹ and L^∞ endpoints, with BMO serving as the substitute for L^∞ at the critical regime. In one dimension, the function log|x| exemplifies these properties as the canonical non-L^∞ element of BMO(ℝ), with ||log|x|||_* ≈ 1 (up to constants). Its Hilbert transform H(log|x|) = -\frac{\pi}{2} \sgn(x) almost everywhere, which is bounded and hence lies in BMO(ℝ), underscoring the preservation of the space under H.^[16] More generally, BMO facilitates solutions to problems involving singular integrals, such as non-tangential boundary limits of harmonic functions in the upper half-plane whose maximal functions lie in L², yielding boundary values in BMO via the Poisson integral and Hilbert transform.

Variants and extensions

Analytic BMO spaces

Analytic BMO spaces consist of holomorphic functions on specific domains that satisfy the bounded mean oscillation condition with respect to suitable subdomains. On the unit disk \mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}, the space BMOA comprises all analytic functions f in \mathbb{D} such that the boundary values f^* (in the sense of nontangential limits) belong to the BMO space on the unit circle \mathbb{T}, equipped with the norm \|f\|_{\mathrm{BMOA}} = \|f^*\|_{\mathrm{BMO}}. This definition ensures that the mean oscillation of f^* over arcs of \mathbb{T} is uniformly bounded: \sup_I \frac{1}{|I|} \int_I |f^*(\mathrm{e}^{i\theta}) - f_I^*| \, d\theta < \infty, where the supremum is over arcs I \subset \mathbb{T} and f_I^* is the average over I. Analogously, on the upper half-plane \mathbb{H} = \{ z \in \mathbb{C} : \operatorname{Im} z > 0 \}, the space BMOH (or analytic BMO on \mathbb{H}) includes holomorphic functions g on \mathbb{H} whose boundary values on \mathbb{R} have bounded mean oscillation, with the norm defined via suprema over intervals J \subset \mathbb{R} of the average deviation: \sup_J \frac{1}{|J|} \int_J |g(x) - g_J| \, dx < \infty. These spaces arise naturally in complex analysis, adapting the real-variable BMO condition to holomorphic settings through boundary behavior. A key feature is the duality: the dual of the analytic Hardy space H^1(\mathbb{D}) is precisely BMOA, under the pairing \langle f, g \rangle = \int_{\mathbb{T}} f(\mathrm{e}^{i\theta}) \overline{g(\mathrm{e}^{i\theta})} \, dm(\theta), where m is normalized Lebesgue measure on \mathbb{T}, and boundary values are taken in the L^1 sense. This duality extends to the half-plane, where the dual of H^1(\mathbb{H}) is BMOH, with pairing via boundary integrals on \mathbb{R}. Such dual pairings highlight the role of analytic BMO spaces in multiplier theory and operator algebras on Hardy spaces. BMOA functions satisfy several important properties. Notably, f \in \mathrm{BMOA} if and only if the measure \mu_f(dz) = |f'(z)|^2 (1 - |z|^2) \, dA(z), where dA is area measure on \mathbb{D}, is a Carleson measure: \sup_I \frac{\mu_f(\gamma(I))}{|I|} < \infty, with \gamma(I) the Carleson square over arc I \subset \mathbb{T}. This characterization links BMOA to embedding theorems for derivatives. Additionally, every function in BMOA belongs to the Nevanlinna class \mathcal{N}, the space of analytic functions f on \mathbb{D} for which \sup_r \int_0^{2\pi} \log^+ |f(re^{i\theta})| \, d\theta < \infty. The space BMOA properly contains the bounded analytic functions H^\infty(\mathbb{D}), as \|f\|_{\mathrm{BMOA}} \leq 2 \|f\|_{L^\infty} for f \in H^\infty, but there exist unbounded examples in BMOA. For instance, finite Blaschke products, such as B(z) = z or B(z) = \frac{z - a}{1 - \overline{a} z} for |a| < 1, lie in BMOA since they are in H^\infty. A classic unbounded example is f(z) = \log(1 - z), whose boundary values have bounded mean oscillation on \mathbb{T} excluding the origin.

Vanishing mean oscillation space

The space of functions of vanishing mean oscillation, denoted VMO(\mathbb{R}^n), consists of those f \in BMO(\mathbb{R}^n) satisfying

\lim_{|Q| \to 0} \sup \frac{1}{|Q|} \int_Q |f(x) - f_Q| \, dx = 0,

where the limit is taken uniformly over all cubes Q \subset \mathbb{R}^n and f_Q denotes the average of f over Q.^[17] Equivalently, VMO(\mathbb{R}^n) is the closure of L^\infty(\mathbb{R}^n) in the BMO norm, or more precisely, the closure of the bounded uniformly continuous functions within BMO(\mathbb{R}^n).^[17] VMO(\mathbb{R}^n) forms a closed subspace of BMO(\mathbb{R}^n).^[17] It contains all continuous functions vanishing at infinity, C_0(\mathbb{R}^n), which is dense in VMO(\mathbb{R}^n) under the BMO norm.^[18] The dual of VMO(\mathbb{R}^n) is the real Hardy space H^1(\mathbb{R}^n), establishing an isometric isomorphism via the duality pairing from BMO(\mathbb{R}^n) and H^1(\mathbb{R}^n). Functions in VMO(\mathbb{R}^n) admit characterizations analogous to those in BMO(\mathbb{R}^n) but with uniformity conditions reflecting the vanishing oscillation at small scales. One such characterization extends the John-Nirenberg inequality: for f \in VMO(\mathbb{R}^n), there exist absolute constants C, c > 0 such that for every cube Q,

|Q|^{-1} |\{ x \in Q : |f(x) - f_Q| > t \}| \leq C \exp\left( -c \frac{t}{\|f\|_{\mathrm{BMO}}} \right)

holds uniformly as |Q| \to 0, with the implied constants independent of Q.^[19] Additionally, elements of VMO(\mathbb{R}^n) possess atomic decompositions where the atoms exhibit vanishing moments of order increasing with the inverse scale, ensuring the oscillation control.^[19] VMO(\mathbb{R}^n) serves as the predual of the space of uniformly continuous singular integral operators on H^1(\mathbb{R}^n).

Dyadic BMO space

The dyadic BMO space on \mathbb{R}^n, often denoted {\rm BMO}_d(\mathbb{R}^n), is defined for locally integrable functions f as the collection of those for which the supremum of the mean oscillation over all dyadic cubes is finite. Dyadic cubes are generated by the grid $2^{-k} \mathbb{Z}^n for k \in \mathbb{Z}, consisting of cubes with side lengths $2^{-k} and vertices at points in this grid. The associated seminorm is given by

\|f\|_{{\rm BMO}_d} = \sup_{Q \in \mathcal{D}} \frac{1}{|Q|} \int_Q |f(x) - f_Q| \, dx,

where \mathcal{D} denotes the family of all dyadic cubes, |Q| is the Lebesgue measure of Q, and f_Q = \frac{1}{|Q|} \int_Q f(x) \, dx is the average of f over Q.^[20] This space forms a Banach space under the norm \|f\|_{{\rm BMO}_d} + \|f\|_{L^\infty_{\rm loc}}, and it is a proper subspace of the full BMO space, with the inclusion continuous in the sense that \|f\|_{{\rm BMO}_d} \leq \|f\|_{{\rm BMO}} for f \in {\rm BMO}_d(\mathbb{R}^n). However, the norms are equivalent up to universal constants: there exist c, C > 0 independent of f such that c \|f\|_{{\rm BMO}} \leq \|f\|_{{\rm BMO}_d} \leq C \|f\|_{{\rm BMO}} when f \in {\rm BMO}_d(\mathbb{R}^n). Moreover, the step functions constant on dyadic cubes are dense in the full BMO space with respect to the BMO norm. The nested structure of dyadic cubes simplifies computations compared to the full BMO space, where averages are taken over arbitrary cubes, while preserving key analytic properties.^[20]^[21]^[22] A fundamental tool in controlling the {\rm BMO}_d norm is the dyadic maximal function, defined as M_d f(x) = \sup_{Q \in \mathcal{D}, x \in Q} |f_Q|, which bounds the oscillations and facilitates estimates in discrete settings. The space {\rm BMO}_d(\mathbb{R}^n) is particularly useful for expansions in the Haar basis, where the coefficients of functions in {\rm BMO}_d satisfy controlled \ell^\infty growth due to the dyadic alignment of Haar wavelets with the cube grid. This connection extends to broader wavelet theory, enabling decompositions and approximations of BMO functions via dyadic systems, which are foundational in multiresolution analysis.^[20]^[23] Unlike the full BMO space, which requires handling translations and rotations of cubes, the dyadic variant allows for more straightforward algorithmic and theoretical treatments, such as in discrete harmonic analysis. Nonetheless, {\rm BMO}_d(\mathbb{R}^n) shares the duality with the dyadic Hardy space {\rm H}^1_d(\mathbb{R}^n), defined via dyadic atoms or the dyadic maximal function, mirroring the classical Fefferman-Stein duality for BMO and {\rm H}^1. This duality underpins applications in operator theory and singular integrals restricted to dyadic grids.^[20]^[21]

Recent generalizations

Recent generalizations of bounded mean oscillation (BMO) spaces have extended the classical framework to accommodate fractional orders, multilinear structures, and multi-parameter settings, addressing applications in nonlocal partial differential equations (PDEs) and higher-dimensional singular integral operators. These developments, primarily from the 2010s onward, fill gaps in classical harmonic analysis by incorporating scaling behaviors and product structures that arise in modern operator theory.^[24] Fractional BMO spaces, denoted BMO^α for 0 < α < 1, modify the oscillation measure by considering averages over balls scaled by radius r^α, capturing sublinear growth relevant to fractional integrals. Specifically, the norm is defined as |b|{BMO{r,\eta}} = \sup_{R \in \mathcal{D}} |R|^\eta \langle |b - b_R| \rangle_{r,R}, where \eta \in [0, m), r \in [1, \infty), \mathcal{D} is a dyadic lattice, and \langle f \rangle_{r,R} denotes the fractional r-average over R. This space admits a Haar decomposition: |b|{BMO{2,\eta}} = \sup_{R \in \mathcal{D}} \left( |R|^{2\eta - 1} \sum_{Q \subseteq R} |\langle b, h_Q \rangle|^2 \right)^{1/2}, linking it to dyadic models. Such spaces have been instrumental in establishing boundedness for multilinear fractional operators on weighted Lebesgue spaces, extending classical John-Nirenberg inequalities to fractional settings.^[24]^[24] Multilinear BMO spaces generalize to tuples (f_1, \dots, f_m), with the norm |(f_1, \dots, f_m)|{BMO} = \sup_Q \prod{i=1}^m \left( \frac{1}{|Q|} \int_Q |f_i - f_{i,Q}| \right)^{1/m} < \infty, where the supremum is over cubes Q and f_{i,Q} is the average of f_i over Q. This structure duality pairs with multilinear Hardy spaces H^1, analogous to the classical case, enabling estimates for commutators of multilinear fractional maximal and integral operators on product generalized Morrey spaces. Recent works establish multilinear BMO estimates for these commutators, confirming boundedness from product vanishing Morrey spaces to multilinear BMO, with applications to sparse domination and weighted inequalities.^[24]^[25]^[25] Mixed λ-central BMO spaces refine central BMO by incorporating a parameter λ < 1/n, defined as |f|{CBMO{\tilde{q},\lambda}(\mathbb{R}^n)} = \sup_{r>0} \frac{|(f - f_{B(0,r)}) \chi_{B(0,r)}|{L^{\tilde{q}}(\mathbb{R}^n)}{|B(0,r)|^\lambda |\chi{B(0,r)}|_{L^{\tilde{q}}(\mathbb{R}^n)}}, for 1 < \tilde{q} < \infty and f \in M(\mathbb{R}^n). These spaces characterize the boundedness of commutators with Hardy-Littlewood maximal operators and their adjoints on mixed λ-central Morrey spaces, providing tools for variable exponent settings in singular integrals.^[26]^[26] Bi-parameter BMO spaces on \mathbb{R}^2 extend to product grids, with norms like |B|_{bmo(U,V,p)} = \sup_R \left( \int_R |V(x)^{1/p} (B(x) - \langle B \rangle_R) U_R^{-1}|^p , dx \right)^{1/p}, over rectangles R and matrix weights U, V in A_p classes. Recent equivalences between such norms facilitate lower bounds for commutators with Riesz transforms and upper bounds for Journé operators, supporting analysis of multi-dimensional oscillatory integrals. These generalizations appear in solutions to fractional Navier-Stokes equations, where fractional BMO controls regularity for nonlocal terms.^[27]^[27]^[28]

Examples and applications

Classical examples

One of the fundamental inclusions in the theory of bounded mean oscillation spaces is that L^\infty(\mathbb{R}^n) \subset \mathrm{BMO}(\mathbb{R}^n), with \|f\|_{\mathrm{BMO}} \leq 2\|f\|_{L^\infty}. This follows directly from the definition, as the mean oscillation of an essentially bounded function over any cube is controlled by twice its essential supremum norm.^[6] A prototypical example of a function in \mathrm{BMO}(\mathbb{R}^n) that lies outside L^\infty(\mathbb{R}^n) is the logarithm f(x) = \log|x| defined on \mathbb{R}^n \setminus \{0\}. This function exhibits bounded mean oscillation with \|f\|_{\mathrm{BMO}(\mathbb{R}^n)} \approx 1, independent of dimension n \geq 1, despite being unbounded near the origin and diverging logarithmically at infinity.^[6] Riesz potentials provide another class of classical examples in \mathrm{BMO}(\mathbb{R}^n). Specifically, for $0 < \alpha < n, the Riesz potential operator I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}} \, dy maps L^{n/\alpha}(\mathbb{R}^n) continuously into \mathrm{BMO}(\mathbb{R}^n). For functions f \in L^1(\mathbb{R}^n) with suitable decay or support conditions, such as compact support, the parameter choice \alpha = n(1 - 1/p) for p > 1 near the endpoint ensures I_\alpha f \in \mathrm{BMO}(\mathbb{R}^n), illustrating the endpoint behavior beyond the standard Hardy-Littlewood-Sobolev inequalities for Lebesgue spaces.^[29] Boundary values of bounded harmonic functions in domains like the unit ball or upper half-space also belong to \mathrm{BMO}. Non-constant examples arise as Poisson kernel integrals of non-constant L^\infty data on the boundary, yielding bounded harmonic extensions whose nontangential boundary limits form non-constant functions in \mathrm{BMO} (via the inclusion L^\infty \subset \mathrm{BMO}). For instance, the Poisson integral of a step function on the circle produces a bounded harmonic function in the disk with boundary values exhibiting controlled oscillation.^[30] Finally, characteristic functions of cubes demonstrate functions in \mathrm{BMO}(\mathbb{R}^n) that fail to belong to the subspace of vanishing mean oscillation (VMO). The characteristic function \chi_Q of a unit cube Q has bounded mean oscillation over larger cubes, but the oscillation does not tend to zero uniformly as the cube size shrinks to zero, placing it outside VMO.^[31]

Applications in analysis

Bounded mean oscillation (BMO) functions play a crucial role in the theory of singular integral operators, particularly those of Calderón-Zygmund type. A fundamental result establishes that such operators map L^\infty continuously into BMO, providing an endpoint substitute for bounded functions in harmonic analysis where L^\infty estimates fail.^[11] This mapping property underpins the boundedness of commutators [b, T]f = T(bf) - bTf on L^p spaces for $1 < p < \infty, when b \in BMO and T is a Calderón-Zygmund operator, as shown in the multiplier theorem framework.^[32] These solvability results extend to multilinear settings, where BMO coefficients ensure operator boundedness on product spaces like Sobolev or Hardy classes.^[33] In partial differential equations (PDEs), BMO estimates provide sharp regularity control for solutions to elliptic and parabolic problems. The De Giorgi-Nash-Moser theory yields Hölder continuity for solutions of uniformly elliptic equations with bounded measurable coefficients, but refined BMO solvability further characterizes the oscillation of solutions or their gradients.^[34] For instance, in divergence-form elliptic operators with A_\infty weights, solutions belong to BMO, enabling uniform-in-time estimates for inhomogeneous parabolic equations like the thermistor problem.^[35] In quasilinear elliptic systems, such as the p-Laplacian, weak solutions have gradients in BMO, facilitating higher-order regularity via iteration techniques.^[36] The Corona theorem highlights BMO's role in approximation theory within the Hardy space H^\infty on the unit ball or disk. Carleson's original result asserts that if \sum_{j=1}^N |f_j| \geq \delta > 0 on the domain with f_j \in H^\infty, then there exist g_j \in H^\infty such that \sum f_j g_j = 1. Extensions show that solutions g_j can be chosen in the analytic BMO space (BMOA), with explicit BMO norms bounding the approximation error relative to the data.^[37] This BMO solvability ensures dense generation of the domain by H^\infty functions under Corona conditions, connecting to Carleson measure characterizations. Recent applications extend BMO to nonlocal operators, such as the fractional Laplacian. For s-fractional p-Laplacian equations in divergence form with BMO kernel coefficients, solutions satisfy W^{s+\sigma,q} estimates for suitable q, improving upon classical local regularity.^[38] These results, emerging in the 2020s, address discontinuous coefficients in nonlocal PDEs modeling diffusion or finance. Additionally, Carleson embeddings leverage BMO duality: a measure \mu on the boundary is Carleson if and only if the embedding H^1 \to L^1(\mu) is bounded, with BMO functions inducing such measures via their Poisson extensions. This framework supports matrix-weighted and multiparameter variants in modern analysis.^[39]

References

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Since Pt ∗ const = const, Pt ∗ f is determined up to constants on equivalence classes in BMO(Rn). f ∈ BMO(Rn) ⇒ |f| is in BMO(Rn) : 1 m(B).
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### Summary of Hilbert Transform and Singular Integrals in Relation to BMO
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