Wavelet expansions for weighted, vector-valued BMO functions (original) (raw)

Characterization of weighted function spaces in terms of wavelet transforms

Boletim da Sociedade Paranaense de Matemática, 2018

In this paper, we have characterized a weighted function space $ B_{\omega,\psi}^{p,q}, ~ 1\leq p,q<\infty$ in terms of wavelet transform and shown that the norms on the spaces Bomega,psip,qB_{\omega,\psi}^{p,q}Bomega,psip,q and bigwedgeomegap,q\bigwedge_\omega^{p,q}bigwedgeomegap,q (the space defined in terms of differences trianglex\triangle_xtrianglex) are equivalent.

Multiresolution Approximations and Wavelet Bases of Weighted L p Spaces

Journal of Fourier Analysis and Applications, 2003

We study boundedness and convergence on L p (R n , dµ) of the projection operators P j given by MRA structures with non-necessarily compactly supported scaling function. As a consequence, we prove that if w is a locally integrable function such that w − 1 p−1 (x)(1 + |x|) −N is integrable for some N > 0, then the Muckenhoupt A p condition is necessary and sufficient for the associated wavelet system to be an unconditional basis for the weighted space L p (R n , w(x) dx), 1 < p < ∞. Math Subject Classifications. 42C15, 42B20. Keywords and Phrases. wavelets, weighted Lebesgue spaces, A p weights. Acknowledgments and Notes. First author was supported by CONICET and Prog. CAID+D-UNL; second author was supported by CONICET, PICT 98 (Código 03-04186) and Prog. CAI+D-UNL, and third author was partially supported by D.G.E.S. grant (PB97-1097), Junta de Andalucía and UNL.

Wavelet characterization of weighted spaces

Journal of Geometric Analysis, 2001

We give a characterization of weighted Hardy spaces H p (w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A ∞We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional basis and a criterion for membership into the space from the size of the wavelet coefficients.

A wavelet characterization for the dual of weighted Hardy spaces

Proceedings of the American Mathematical Society, 2009

We define the weighted Carleson measure space CM O p w using wavelets, where the weight function w belongs to the Muckenhoupt class. Then we show that CM O p w is the dual space of the weighted Hardy space H p w by using sequence spaces. As an application, we give a wavelet characterization of BM O w .

New Classes of Weighted Hölder-Zygmund Spaces and the Wavelet Transform

Journal of Function Spaces and Applications, 2012

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces𝒮0(ℝn)and𝒮(ℍn+1). We then introduce and study a new class of weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.

ASYMPTOTIC BERNSTEIN TYPE INEQUALITIES AND ESTIMATION OF WAVELET COEFFICIENTS

In this paper, we investigate the wavelet coefficients for function spaces A p k := {f : (iω) k ˆ f (ω)p 1}, k ∈ N ∪ {0}, p ∈ (1, ∞) using an important quantity C k,p (ψ) := sup{ |f, ψ|ˆψp : f ∈ A p ′ k } with 1/p + 1/p ′ = 1. In particular, Bernstein type inequalities associated with wavelets are established. We obtained an sharp inequality of Bernstein type for splines and a lower bound for the quantity C k,p (ψ) with ψ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. Comparison of these two families is done by using the quantity C k,p (ψ).

Greedy wavelet projections are bounded on BV

2007

Let BV = BV(R d ) be the space of functions of bounded variation on R d with d ≥ 2. Let ψ λ , λ ∈ ∆, be a wavelet system of compactly supported functions normalized in BV, i.e.,

Explicit Bernstein Type Inequalities and Estimation of Wavelet Coefficients

In this paper, we investigate the wavelet coefficients for function spaces A p k := { f : ∥(iω) k ˆ f (ω)∥ p 1}, k ∈ N ∪ {0}, p ∈ (1, ∞) using an important quantity C k,p (ψ) := sup{ ⟨ f,ψ⟩ ∥ ˆ ψ∥ p : f ∈ A p ′ k } with 1/p + 1/p ′ = 1. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines and a lower bound for the quantity C k,p (ψ) with ψ being the semiorthog-onal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogo-nal spline wavelets. Comparison of these two families is done by using the quantity C k,p (ψ).