On BMO-regular couples of lattices of measurable functions

S. V. Kislyakov¹

¹ Russian Academy of Sciences Steklov Mathematical Institute St. Petersburg Department Fontanka 27 St. Petersburg 191023, Russia

Studia Mathematica, Tome 159 (2003) no. 2, pp. 277-290

Résumé

We introduce a new “weak” BMO-regularity condition for couples $(X,Y)$ of lattices of measurable functions on the circle (Definition 3, Section 9), describe it in terms of the lattice $X^{1/2}(Y')^{1/2}$, and prove that this condition still ensures “good” interpolation for the couple $(X_A,Y_A)$ of the Hardy-type spaces corresponding to $X$ and $Y$ (Theorem 1, Section 9). Also, we present a neat version of Pisier's approach to interpolation of Hardy-type subspaces (Theorem 2, Section 13). These two main results of the paper are proved in Sections 10–18, where some related material of independent interest is also discussed. Sections 1–8 are devoted to the background and motivations, and also include a short survey of some previously known results concerning BMO-regularity. To a certain extent, the layout of the paper models that of the lecture delivered by the author at the conference in functional analysis in honour of Aleksander Pełczyński (Bedlewo, September 22–29, 2002).

Détail

DOI : 10.4064/sm159-2-8

Keywords: introduce weak bmo regularity condition couples lattices measurable functions circle definition section describe terms lattice prove condition still ensures interpolation couple a hardy type spaces corresponding theorem section present neat version pisiers approach interpolation hardy type subspaces theorem section these main results paper proved sections where related material independent interest discussed sections devoted background motivations include short survey previously known results concerning bmo regularity certain extent layout paper models lecture delivered author conference functional analysis honour aleksander czy ski bedlewo september