The Banach space \(H^ 1 (X, d,\mu)\). II
We give the isomorphic classification of atomic \(H^1(X, d, \mu)\), where \((X, d, \mu)\) is a space of homogeneous type, hereby completing a line of investigation opened by the work of B. Maurey and continued by L. Carleson and P. Wojtaszczyk. The resulting isomorphic representatives are dyadic \(H^1\), \((\sum H^1_n)_{\ell^1}\) and \(\ell^1\); each isomorphic type is characterized by geometric properties of \((X, d, \mu)\). Technically, the present paper deals with the existence of ``Franklin- type'' functions on a space of homogeneous type \((X, d, \mu)\) and with the boundedness (in \(H^1(X, d, \mu)\)) of the orthogonal projection onto the span of this ``Franklin-system''. To this end a construction of S. Jaffard and Y. Meyer, together with methods of P. Wojtaszczyk have been adapted. Applying Wojtaszczyk's method one concludes that the ``Franklin-system'' in \(H^1(X, d, \mu)\) is equivalent to special three valued martingale differences in some martingale \(H^1\) space. Using repeatedly Pełczyński's decomposition method, this allows us to reduce in Section 5 the classification problem for atomic \(H^1\) spaces to some of the author's previous work on this subject: -- Classification of (the span of) special three valued martingale differences in martingale \(H^1\) space. -- Classification of the isomorphic type of martingale \(H^1\) space. -- Atomic \(H^1\) spaces are isomorphic to complemented subspaces of martingale \(H^1\). [For part I see the review above].
isomorphic classification of atomic \(H^ 1(X, d, \mu)\)
three valued martingale differences in martingale \(H^ 1\) space
scientific article; zbMATH DE number 840198