Wavelets on self-similar sets and the structure of the spaces \(M^{1,p}(E,\mu)\) (Q2778318)

This thesis studies properties of the space \(M^{1,p}(E, \mu)\), \(1 < p < \infty\), which is defined as in [\textit{P. Hajlasz}, Potential Anal. 5, No. 4, 403-415 (1996; Zbl 0859.46022)]. In that paper it was shown that \(M^{1,p}(E, \mu)\) is a Sobolev and Banach space. In the article under review the set \(E\) is assumed to be self-similar and of Cantor type, and \(\mu\) is assumed to be a natural invariant measure associated to \(E\). Using wavelet analysis tools the author shows that with these restrictions on \(E\) and \(\mu\), the space \(M^{1,p}(E, \mu)\) is not reflexive. He also studies which functions \(\psi\) qualify as a wavelet associated with a multiresolution analysis on self-similar sets. In particular, he shows that the wavelet bases for \(L^2(E, \mu)\) are unconditional bases for \(L^p(E, \mu)\), \(1 < p < \infty\). This generalizes work of \textit{A. Jonsson} [J. Fourier Anal. Appl. 4, No. 3, 329-340 (1998; Zbl 0912.42025)], where wavelet bases were obtained for sets satisfying the so-called Markov's inequality. The proof uses Calderón-Zygmund theory.