We study the distance in the Zygmund class Λ* to the subspace I(BMO) of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of I(BMO) in the Zygmund seminorm. We also generalise this result to Zygmund measures on Rd. Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in Λ* that are also in the classical Sobolev space W1,p, for 1
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.
It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. As an application, we use them to estimate the size of irregular points of inner functions omitting large sets. Finally, we also present a natural interpretation of the results in the upper half plane.
Ó. Blasco and S. Pott showed that the supremum of operator norms over L2$L^2$ of all bicommutators (with the same symbol) of one‐parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1
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