Function spaces on the Koch curve (Q614187)

Two Besov classes, \({\mathbf B}^s_p\) and \({\mathcal B}^s_p\) (\(0<s<1\), \(1<p<\infty\)), on the Koch curve (which is a \(d\)-set with \(d=\log 4/\log 3\)) are defined by the trace method and using the snowflake transform, respectively. Using a wavelet characterization, the author shows that \({\mathbf B}^s_p\) and \({\mathcal B}^s_p\) are isomorphic, via the wavelet coefficients, to the spaces of sequences \(\{ a_w,b_w\}\) such that \[ \sum_{w\in W_L} |a_w|^p +\sum_{j=0}^\infty 2^{j(sp-1)}\sum_{w\in W_{j+L}} |b_w|^p<\infty \] and \[ \sum_{w\in W_L} |a_w|^p +\sum_{j=0}^\infty (\sqrt 3)^{j(sp-1)}\sum_{w\in W_{j+L}} |b_w|^p<\infty \] respectively. Here \(W_m\) stands for words with length \(m\) and \(L\) depends on the wavelet system.