INTERPOLATION OF WEIGHTED BANACH LATTICES
9
Some of the results presented here were announced in [CN2].
(b) The contents of each section of this paper.
In Section 1 we fix definitions and terminology for Banach lattices of measurable
functions and for interpolation theory. We also give some preliminary results. Some
of these deal with the interaction between the Fatou property, cr-order continuity and
interpolation methods, including elementary ones such as Gagliardo completions, sums
and intersections. We observe that the formulas (XoDXiY =
XQ+X[
and
(XQ-\-XIY
=
XQ n X[ hold for associate spaces, analogously to the well known formulae for dual
spaces. We have been informed that Lozanovskii has also obtained these formulae. We
discuss the continuity of complex interpolation norms ||x||[x]e as functions of 0. We also
note that any saturated Banach lattice of measurable functions is "locally" a Kothe
function space. We would expect at least some of these results to be already known,
but we have not found any reference to them in the literature.
Section 2 contains full statements of our main theorems, which we have already
sketched above. It also includes the result which enables them to be extended to lattices
which are not saturated.
Section 3 is devoted to a uniqueness theorem for "Calderon products" i.e. lattices
XQ~ Xf which are generated from a couple of Banach lattices XQ and X\ via a con-
struction defined in [Cal] and which often coincide with complex interpolation spaces.
We show that, under appropriate hypotheses, if XQ~ Xf = YQ~ Yf for at least two
distinct values of 0 in [0,1] then Xo = YQ and X\ —Y\. We also briefly treat some
analogous partial results for the real interpolation method. The reader can guess of
course that we need this uniqueness theorem to treat the special case of our main result
when X = Y.
Section 4 studies two deeper properties of the ^-functional for a couple of Banach
lattices of measurable functions. Our first result states that the K-functional of an
element x with respect to such a couple X =
(XQ,X\)
is always equivalent to an
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