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Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
F. John and L. Nirenberg [JN1961] introduced the space of functions of bounded mean oscillation (abbreviated BMO) and they applied it to smoothness problems in partial differential equations. Their results show that BMO functions cannot become unbounded too rapidly and that BMO is situated between L∞ and Lp for all 0 < p < ∞. Later on, C. Fefferman and E. Stein [FS1972] obtained the principal result on the connection between BMO and conjugate harmonic functions. This concludes that BMO consists exactly of all sums of pairs of analytic functions, one of which has a bounded real part and another one has a bounded imaginary part. C. Fefferman’s famous equation, (H1)∗ = BMOA, describes a deep relation between BMO and the Hardy space [Fe1971]. Here BMOA denotes the set of analytic functions f(z) in the unit disc D on the complex plane ℂ whose boundary function f (eiθ) belongs to BMO of the unit circle, that is, () ‖f‖BMO:=supI1|I|∫I|f(eiθ)−1|I|∫If(eit)dt|dθ<∞,
Applications of Singular Integral Operators and Commutators
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
BMO, the space of functions having bounded mean oscillation, was introduced by Fritz John and Louis Nirenberg in 1961 (see John and Nirenberg (1961)) in view of the mathematics community’s interest in real analysis and in studies of partial differential equations.
On the continuation principle of local smooth solution for the Hall-MHD equations
Published in Applicable Analysis, 2022
Ravi P. Agarwal, Ahmad M. A. Alghamdi, Sadek Gala, Maria Alessandra Ragusa
In [21] (see also [22]), Wang and Li obtained a Beale–Kato–Majda-type criteria of smooth solutions based on homogeneous Besov norms. More precisely, they showed that a solution with -data, , is a smooth solution provided that Here and hereafter, stands for the homogenous Besov space. For the definition of Besov spaces, see [23]. Their proof is based on the logarithmic Sobolev inequality in the Besov spaces which need the -norm, and by using the property of Hardy and BMO spaces. Here, BMO denotes the space of functions of bounded mean oscillations.