Multivariate wavelet leaders Rényi dimension and multifractal formalism in mixed Besov spaces

2021 ◽  
Author(s):  
Moez Ben Abid ◽  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Maamoun Turkawi
Keyword(s):  
Lower Bound ◽  
Besov Space ◽  
Besov Spaces ◽  
Multifractal Formalism ◽  
Wavelet Leaders ◽  
Rényi Dimension ◽  
Generic Sense

In this paper, we first establish a general lower bound for the multivariate wavelet leaders Rényi dimension valid for any pair [Formula: see text] of functions on [Formula: see text] where [Formula: see text] belongs to the Besov space [Formula: see text] with [Formula: see text] and [Formula: see text] belongs to [Formula: see text] with [Formula: see text]. We then prove the optimality of this result for quasi all pairs [Formula: see text] in the Baire generic sense. Finally, we compute both iso-mixed and upper-multivariate Hölder spectra for all pairs [Formula: see text] in the same [Formula: see text]-set. This allows to prove (respectively, study) the Baire generic validity of the upper-multivariate (respectively, iso-multivariate) multifractal formalism based on wavelet leaders for such pairs.

scholarly journals On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Moez Ben Abid ◽  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Borhen Halouani
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Besov Space ◽  
Sobolev Spaces ◽  
Wavelet Coefficients ◽  
Multifractal Formalism ◽  
Spectrum Of A Function ◽  
Set Of Points

The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.

Mixed Wavelet Leaders Multifractal Formalism in a Product of Critical Besov Spaces

2017 ◽  
Vol 14 (4) ◽  
Author(s):  
Moez Ben Abid ◽  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Borhen Halouani
Keyword(s):  
Besov Spaces ◽  
Multifractal Formalism ◽  
Wavelet Leaders ◽  
Critical Besov Spaces

scholarly journals Extended Cesáro operators between generalized Besov spaces and Bloch type spaces in the unit ball

2009 ◽  
Vol 7 (3) ◽  
pp. 209-223 ◽  
Author(s):  
Ze-Hua Zhou ◽  
Min Zhu
Keyword(s):  
Unit Ball ◽  
Besov Space ◽  
Besov Spaces ◽  
Complex Space ◽  
Bloch Space ◽  
Sufficient Conditions ◽  
Cesàro Operator ◽  
Type Spaces ◽  
Necessary And Sufficient ◽  
Cesaro Operator

Let 𝑔 be a holomorphic of the unit ballBin then-dimensional complex space, and denote byTgthe extended Cesáro operator with symbolg. Let 0 <p< +∞, −n− 1 <q< +∞,q> −1 and α > 0, starting with a brief introduction to well known results about Cesáro operator, we investigate the boundedness and compactness ofTgbetween generalized Besov spaceB(p, q)and 𝛼α- Bloch spaceℬαin the unit ball, and also present some necessary and sufficient conditions.

scholarly journals Weighted holomorphic Besov spaces on the polydisk

2011 ◽  
Vol 9 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Anahit V. Harutyunyan ◽  
Wolfgang Lusky
Keyword(s):  
Fractional Derivative ◽  
Besov Space ◽  
Besov Spaces ◽  
Regular Variation ◽  
Holomorphic Functions ◽  
Spaces Of Holomorphic Functions ◽  
Weighted Besov Spaces ◽  
Unit Polydisk ◽  
Projection Theorems ◽  
Dimensional Lebesgue Measure

This work is an introduction of weighted Besov spaces of holomorphic functions on the polydisk. LetUnbe the unit polydisk inCnandSbe the space of functions of regular variation. Let1≤p<∞,ω=(ω1,…,ωn),ωj∈S(1≤j≤n)andf∈H(Un).The functionfis said to be an element of the holomorphic Besov spaceBp(ω)if‖f‖Bp(ω)p=∫Un|Df(z)|p∏j=1nωj(1-|zj|)/(1-|zj|2)2-pdm2n(z)<+∞, wheredm2n(z)is the2n-dimensional Lebesgue measure onUnandDstands for a special fractional derivative offdefined in the paper. For example, ifn=1thenDfis the derivative of the functionzf(z).We describe the holomorphic Besov space in terms ofLp(ω)space. Moreover projection theorems and theorems of the existence of a right inverse are proved.

On the continuity of pseudo-differential operators on Besov spaces

Analysis ◽  
2006 ◽  
Vol 26 (4) ◽  
Author(s):  
Madani Moussai
Keyword(s):  
Besov Space ◽  
Besov Spaces ◽  
Differential Operators ◽  
Positive Real ◽  
Pseudo Differential Operators

We study the continuity of pseudo-differential operators on Besov spacefor some positive real

scholarly journals Motion Groups and Absolutely Convergent Fourier Transforms

1987 ◽  
Vol 43 (3) ◽  
pp. 385-397 ◽  
Author(s):  
Garthi Gaudry ◽  
Rita Pini
Keyword(s):  
Fourier Transform ◽  
Besov Space ◽  
Besov Spaces ◽  
Fourier Transforms ◽  
Classical Theorem ◽  
Motion Groups

According to an extension of a classical theorem of Bernstein, due to C. Herz, a function on Rn belonging to a Besov space of appropriate order has an absolutely convergent Fourier transform. We establish extensions of this result to Cartan motion groups, for Besov spaces defined with respect to both isotropic and non-isotropic differences.

scholarly journals The Periodic Boundary Value Problem for a Quasilinear Evolution Equation in Besov Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Yun Wu ◽  
Zhengrong Liu ◽  
Xiang Zhang
Keyword(s):  
Evolution Equation ◽  
Boundary Value Problem ◽  
Besov Space ◽  
Besov Spaces ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Value Problem ◽  
Solution Map ◽  
Well Posed

This paper is concerned with the periodic boundary value problem for a quasilinear evolution equation of the following type:∂tu+f(u)∂xu+F(u)=0,x∈T=R/2πZ,t∈R+. Under some conditions, we prove that this equation is locally well-posed in Besov spaceBp,rs(T). Furthermore, we study the continuity of the solution map for this equation inB2,rs(T). Our work improves some earlier results.

Lipschitz and Besov spaces in quantum calculus

2016 ◽  
Vol 19 (03) ◽  
pp. 1650021
Author(s):  
Akram Nemri ◽  
Belgacem Selmi
Keyword(s):  
Harmonic Analysis ◽  
Besov Space ◽  
Besov Spaces ◽  
Time Scale ◽  
Poisson Kernel ◽  
Quantum Calculus ◽  
Weighted Besov Spaces

The purpose of this paper is to investigate the harmonic analysis on the time scale [Formula: see text], [Formula: see text] to introduce [Formula: see text]-weighted Besov spaces subspaces of [Formula: see text] generalizing the classical one. Further, using an example of [Formula: see text]-weighted [Formula: see text] which is introduced and studied. We give a new characterization of the [Formula: see text]-Besov space using [Formula: see text]-Poisson kernel and the [Formula: see text] Littlewood–Paley operator.

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