Approximation and Extension of Functions of Vanishing Mean Oscillation

Vanishing Mean Oscillation and Continuity of Rearrangements

“…For nice domains Ω, VMO(Ω) is the closure, in the BMO-seminorm, of the set of uniformly continuous functions in BMO(Ω) (see [9]). General functions in VMO, though, need be neither continuous nor bounded; an example is (− log |x|) p + for 0 < p < 1.…”

“…On Ω = R n , VMO can also be characterized as the subset of BMO(R n ) on which translation is continuous. In the case when Ω is unbounded, note that there is a strictly smaller VMO-space, sometimes denoted CMO (see [4,9,25]), in which additional vanishing mean oscillation conditions are required as the cube or its sidelength go to infinity.…”

A Survey on Function Spaces of John--Nirenberg Type

“…Iwaniec [52] used the compactness theorem in Uchiyama [99] to study linear complex Beltrami equations and the L p (C)-theory of quasiregular mappings. All these classical results have wide generalizations as well as applications, and inspire a myriad of further studies in recent years; see, for instance, the references [54,25,11,9] for their applications in singular integral operators as well as their commutators, the references [75,74,76,73,72,63] for their applications in pointwise multipliers, the references [24,78,89] for their applications in partial differential equations, and the references [23,6,18,31,32,33] for more variants and properties of BMO (R n ). In particular, we refer the reader to Chang and Sadosky [26] for an instructive survey on functions of bounded mean oscillation, and also Chang et al [23] for BMO spaces on the Lipschitz domain of R n .…”

A Survey on Function Spaces of John–Nirenberg Type

“…For instance, Fefferman and Stein [2] proved that the dual space of the Hardy space H 1 (R n ) is BMO (R n ); Coifman et al [3] showed an equivalent characterization of the boundedness of Calderón-Zygmund commutators via BMO (R n ); Coifman and Weiss [4,5] introduced the space of homogeneous type and studied the Hardy space and the BMO space in this context; Sarason [6] obtained the equivalent characterization of VMO (R n ), the closure in BMO (R n ) of uniformly continuous functions, and used it to study stationary stochastic processes satisfying the strong mixing condition and the algebra H ∞ + C; Uchiyama [7] established an equivalent characterization of the compactness of Calderón-Zygmund commutators via CMO (R n ) which is defined to be the closure in BMO (R n ) of infinitely differentiable functions on R n with compact support; Nakai and Yabuta [8] studied pointwise multipliers for functions on R n of bounded mean oscillation; and Iwaniec [9] used the compactness theorem in Uchiyama [7] to study linear complex Beltrami equations and the L p (C) theory of quasiregular mappings. All these classical results have wide generalizations as well as applications and have inspired a myriad of further studies in recent years: see, for instance, the References [10][11][12][13] for their applications in singular integral operators as well as their commutators, the References [14][15][16][17][18][19] for their applications in pointwise multipliers, the References [20][21][22] for their applications in partial differential equations, and the References [23][24][25][26][27][28] for more variants and properties of BMO (R n ). In particular, we refer the reader to Chang and Sadosky [29] for an instructive survey on functions of bounded mean oscillation and also Chang et al [25] for BMO spaces on the Lipschitz domain of R n .…”