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

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.


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
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