Approximation in the Zygmund and Hölder classes on

Abstract

We determine the distance (up to a multiplicative constant) in the Zygmund class Λ * (R ) to the subspace J (bmo) (R ). The latter space is the image under the Bessel potential(1−Δ) −1/2 of the space bmo(R ), which is a non-homogeneous version of the classical BMO. Locally, J (bmo) (R ) consists of functions that together with their first derivatives are in bmo(R ). More generally, we consider the same question when the Zygmund class is replaced by the Hölder space Λ (R ), with 0 < ≤ 1 and the corresponding subspace is J (bmo) (R ), the image under (1 − Δ) − /2 of bmo(R ). One should note here that Λ 1 (R ) = Λ * (R ). Such results were known earlier only for = = 1 with a proof that does not extend to the general case. Our results are expressed in terms of second differences. As a byproduct of our wavelet based proof, we also obtain the distance from ∈ Λ (R ) to J (bmo) (R ) in terms of the wavelet coefficients of . We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of on the upper half-space R +1 + .MCS class: 26B35.

Bloch functions and Bekollé-Bonami weights

“…The problem of characterizing the closure of H ∞ in B was initially posed in 1974 by Anderson, Clunie and Pommerenke (see [3]) and remains open to this date. It is worth mentioning that several variants of this problem have been studied in [9], [10], [13], [17] and [19]. For 0 < p < ∞, let H p denote the classical Hardy of analytic functions f in D, satisfying…”

“…for any t * > max{t * 0 , 2C(w)} =: t * 1 . Consider the extension f * of f given by (19). The estimate (28) gives that there exists a constant C 2 > 0, such that for any square R ⊂ R 2 , we have…”