Harmonic analysis of the space BV. (English) Zbl 1044.42028

Cohen, Albert; Dahmen, Wolfgang; Daubechies, Ingrid; DeVore, Ronald

Rev. Mat. Iberoam. 19, No. 1, 235-263 (2003).

A function \(f \in L^1(\Omega)\), \(\Omega \subset R^d\), has bounded variation if \[ | f| _{\text{BV}(\Omega)} := \sup_{g\in C_c^1(\Omega, R^d), \| g\| _\infty \leq 1} \int_\Omega f\text{ div}(g) < \infty. \] The space \(\text{BV}(\Omega)\) of such functions is a Banach space when equipped with the norm: \[ \| f\| _{\text{BV}(\Omega)} = \| f\| _1+| f| _{\text{BV}(\Omega)}. \] Let \(D\) be the family of dyadic cubes in \(R^d\). Let \(\psi \in L^2(R^d)\) be a wavelet on \(R\). If \(E\) denotes the set of non-zero vertices of the unit cube in \(R^d\), and if \(e \in E\), then we write \(\psi^e = \psi^{e_1}(x_1)\dots \psi^{e_d}(x_d)\). For \(Q = 2^{-j}(k+[0,1]^d) \in D\) we define \(\psi_Q^e\) to be the wavelet \(\psi^e\) scaled relative to \(Q\). Let \(\widetilde{\psi^e}\) be the dual wavelet. For a tempered distribution \(f\) we define \(f_Q\) to be the vector of coefficients \(\langle f, \widetilde{\psi^e_Q} \rangle\). The paper under review provides the answer to the question: which values of \(\gamma\) satisfy the following inequality: \[ \sum_{Q \in D: | f_Q| > \epsilon | Q| ^\gamma} | Q| ^\gamma \leq C \epsilon^{-1} | f| _{\text{BV}(\Omega)}? \] Applications of the characterization of all such \(\gamma\)’s include almost characterizations of \(\text{BV}\) spaces, Gagliardo-Nirenberg-type inequalities and interpolation spaces between BV and Sobolev or Besov spaces.

Reviewer: Wojciech Czaja (Wrocław)

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
46B70	Interpolation between normed linear spaces
26B35	Special properties of functions of several variables, Hölder conditions, etc.
42B25	Maximal functions, Littlewood-Paley theory
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

bounded variation; wavelet decompositions; weak \(l_1\); \(K\)-functionals; interpolation; Gagliardo-Nirenberg inequalities; Besov spaces

References:

[1]	Bergh, J. and Löfström, J.: Interpolation spaces. Springer Verlag, 1976. · Zbl 0344.46071
[2]	Cohen, A.: Wavelet methods in numerical analysis, in the Handbook of Numerical Analysis, vol. VII, P.-G. Ciarlet et J.-L. Lions eds., Elsevier, Amsterdam, 2000. · Zbl 0976.65124
[3]	Cohen, A., DeVore, R. and Hochmuth, R.: Restricted Nonlinear Approximation. Constructive Approximation 16 (2000), 85-113. · Zbl 0947.41006 · doi:10.1007/s003659910004
[4]	Cohen, A., DeVore, R., Petrushev, P. and Xu, H.: Non linear approximation and the space BV (R2). Amer. J. Math. 121 (1999), 587-628. · Zbl 0931.41019 · doi:10.1353/ajm.1999.0016
[5]	Cohen, A., Meyer, Y. and Oru, F.: Improved Sobolev inequalities. Proceedings séminaires X-EDP, Ecole Polytechnique, Palaiseau, 1998. · Zbl 1069.46503
[6]	Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numerica 6, Cambridge University Press, 1997, 55-228. · Zbl 0884.65106
[7]	Daubechies, I.: Ten Lectures on Wavelets. SIAM, 1992. · Zbl 0776.42018
[8]	Donoho, D., Vetterli, M., DeVore, R. and Daubechies, I.: Har- monic analysis and signal processing. IEEE Trans. Inf. Theory 44 (1998), 2435-2476. · Zbl 1125.94311 · doi:10.1109/18.720544
[9]	DeVore, R.: Nonlinear Approximation. Acta Numerica 7, Cambridge University Press, 1998, 51-150. · Zbl 0931.65007
[10]	DeVore, R. and Petrova, G.: The averaging lemma. J. Amer. Math. Soc. 14 (2000), 279-296. · Zbl 1001.35079 · doi:10.1090/S0894-0347-00-00359-3
[11]	Donoho, D.: Unconditional bases are optimal for data compression and statistical estimation. Appl. Comp. Harm. Anal. 1 (1993), 100-105. · Zbl 0796.62083 · doi:10.1006/acha.1993.1008
[12]	Evans, L. and Gariepy, R.: Measure theory and fine properties of func- tions. CRC Press, New York, 1992. · Zbl 0804.28001
[13]	Meyer, Y.: Ondelettes et Opérateurs. Hermann, Paris, 1990.
[14]	Ziemer, W.: Weakly differentiable functions. Springer Verlag, New York, 1989. · Zbl 0692.46022