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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 18/2002

S. V. Kislyakov
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ON BMO-REGULAR COUPLES OF LATTICES OF MEASURABLE FUNCTIONS

This preprint was accepted October 10, 2002

Contact:
` S. V. Kislyakov `

ABSTRACT:
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 12). 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 Pe\l czy\'nski (B\c
endlewo, September 22--29, 2002).

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