Isomorphisms Between H¹ Spaces
The theory of Hardy spaces is ?rmly rooted in the ?elds of complex analysis, trigonometricseries, and probability. Itshistory began inthedecadebetween1906 and 1916. In 1906 appeared P. Fatou’s treatise on power series and trigonometric series in Acta Mathematica. G. H. Hardy proved in 1915 that integral means of power series in the unit disk are log-convex, just in time to be included in the ?rst edition of E. Landau’s “Darstellung und Begrundung ¨ einiger neuerer Ergebnisse der Funktionentheorie” (1916). In the same year F. and M. Riesz presented their ¨ treatise “Uber die Randwerte einer analytischen Funktion” to the Scandinavian congress of mathematicians in Stockholm. TheclassicaltheoryofHardyspacesisconcernedwiththeboundarybehavior of analytic functions on the unit disk and with estimates for the Fourier co- n cients of the limiting function. Given a power series f(z)= a z, Hardy and n 1 Littlewood (1930) proved that on the unit circle the L norm of its non-tangential 1 maximal function is equivalent to the L norm of its boundary values. R. E. A. C. Paley (1933) shows that the sequence of lacunary coe?cients is square summable provided that the boundary values are integrable on the unit circle, more precisely that ? 1/2 1 2 it a n ? f(e ) dt.
Study of dyadic H^1, its isomorphic invariants and its position within the two classes of martingale and atomic H^1 spaces, and simultaneously, a detailed analysis of the Haar systemOnly basic knowledge in real, complex and functional analysis required, and some probability theory