Differentiability of functions in the Zygmund class

Donaire, Juan Jesús; Llorente, José G.; Nicolau, Artur

doi:10.1112/plms/pdt016

Cited by 10 publications

(8 citation statements)

References 15 publications

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“…Here and hereafter, Dim denotes Hausdorff dimension.The authors asked whether D(f ) (respectively D 0 (f )) should also have maximal Hausdorff dimension if f ∈ Λ * (R d ) ( respectively f ∈ λ * (R d )). In a previous work ( [7]) they showed that this is not the case: the right dimension is 1 and this is the best that can be said in general.…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions

Donaire¹,

Llorente²,

Nicolau³

2014

Rev. Mat. Iberoam.

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Section: Introduction and Main Resultsmentioning

confidence: 95%

Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions

Donaire¹,

Llorente²,

Nicolau³

2014

Rev. Mat. Iberoam.

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“…Reimann [27] showed that for functions satisfying (Q) the induced flows are quasi-symmetric mappings -unfortunately -quasi-symmetric mappings are not necessarily absolutely continuous in R and a function satisfying (Z) needs not be absolutely continuous (cf. [3,27] and [14]). In view of this, some more restrictions on b seem to be necessary for the generalized flow to have an A ∞ density.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Flow With Density and Transport Equation In

Jiang

Xiao

2019

Forum of Mathematics, Sigma

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We show that, if b ∈ L 1 (0, T ; L 1 loc (R)) has spatial derivative in the John-Nirenberg space BMO(R), then it generalizes a unique flow φ(t, ·) which has an A ∞ (R) density for each time t ∈ [0, T ]. Our condition on the map b is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in BMO(R). Statement of main resultsGiven an integer n ≥ 1, a real T ≥ t > 0 and an evolutionary self-map b(t, ·) of R n with b ∈ L 1 (0, T ; L 1 loc (R n )), consider the flow φ(t, x) = x + t 0

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“…The classes Λ have numerous applications, e.g. to PDE's and polynomial approximation or Calderón-Zygmund theory, and have been extensively studied (see for instance [Zyg45], [Ste71, Chapter V], [Tri10], [Mak89], [DLN14]).…”

Section: Introductionmentioning

confidence: 99%

Approximation in the Zygmund and Hölder classes on

Saksman¹,

Gibert

2021

Can. J. Math.-J. Can. Math.

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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.

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Differentiability of functions in the Zygmund class

Cited by 10 publications

References 15 publications

Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions

Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions

Flow With Density and Transport Equation In

Approximation in the Zygmund and Hölder classes on

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