Orthogonal projections on martingale \(H^ 1\) spaces of two parameters
Given natural numbers \(n\) and \(i\) so that \(1\leq i\leq 2^ n\), let \((n, i)\) denote the ``dyadic interval'' \(\left[{i- 1\over 2^ n}, {i\over 2^ n}\right]\), \(h_{(n, i)}\) the \(L^ \infty\)-normalized Haar function supported on \((n, i)\), and \(h_{(n, i)\times (m, j)}(s, t)= h_{(n, i)}(s) h_{(m, j)}(t)\) (for \(s,t\in [0, 1]\)). If \(F= \sum c_{\sigma\times\rho} h_{\sigma\times\rho}\) (for \(\sigma= (n, i)\), \(\rho= (m, j)\)), let \(S(F)= [\sum c^ 2_{\sigma\times \rho}| h_{\sigma\times \rho}|^ 2]^{1/2}\) and define \(H^ 1(\delta^ 2)= \{F\in L^ 1(U)\mid\int_ U S(F)<+ \infty\}\) (where \(U= [0,1]\times [0, 1]\)). Conditions are considered which guarantee that if \(A_ K= (n_ k, i_ k)\times (m_ k, j_ k)\) is a ``dyadic rectangle'' for each \(K\) and the set \(\{A_ K\}\) is pairwise disjoint then the projection \(Pf= \sum_ K (f, \phi_ K)\phi_ K\) is a bounded linear operator on \(H^ 1(\delta^ 2)\), where \(\phi_ K= {\sum_{I\times J\subset A_ K} h_{I\times J}\over \left\| \sum_{I\times J\subset A_ K} h_{I\times J}\right\|_{L^ 2}}\) for all \(K\). The results are highly technical in nature and will not be repeated here. A result analogous to one of \textit{P. W. Jones} [Am. J. Math. 107, 853-893 (1985; Zbl 0595.46017)] for the one-dimensional case is proved and then used to deduce that the space \(H^ 1(\delta^ 2)\) is primary.
scientific article; zbMATH DE number 622558
scientific article