Positivity 10 (2006), 409–429
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/030409-21, published online May 24, 2006
Interpolation of Bilinear Operators Between
and Mieczyslaw Mastylo
Abstract. We study interpolation, generated by an abstract method of means,
of bilinear operators between quasi-Banach spaces. It is shown that under suit-
able conditions on the type of these spaces and the boundedness of the clas-
sical convolution operator between the corresponding quasi-Banach sequence
spaces, bilinear interpolation is possible. Applications to the classical real
method spaces, Calder´on-Lozanovsky spaces, and Lorentz-Zygmund spaces
Mathematics Subject Classiﬁcation (2000). 46B70; 46M35.
Keywords. Interpolation, quasi-Banach spaces, multilinear interpolation,
bilinear operators, Calder´on-Lozanovsky spaces, Lorentz-Zygmund spaces.
Motivated by applications in harmonic analysis, we are interested in interpolation
of bilinear operators deﬁned on products of quasi-Banach spaces. The main aim
of this paper is to prove interpolation theorems for bilinear operators on quasi-
Banach spaces generated by certain interpolation methods. We study a problem
for the abstract method of means as well as for the real interpolation method.
Let us brieﬂy outline the content of the paper. In Section 2 we establish nota-
tion and recall basic facts concerning quasi-Banach spaces and interpolation. In
Section 3 we introduce a notion of special type of convexity for bilinear operators
between quasi-Banach couples and we prove a bilinear interpolation theorem using
the method of means, under the condition that the associated convolution operator
is bounded on the parameter spaces involved in the construction of these meth-
ods. In view of a remarkable result of Kalton , the convexity parameters of the
bilinear operators that take values in so called natural quasi-Banach spaces, are
nicely determined by the types of the domains of the quasi-Banach spaces. We also
The author is supported by the National Science Foundation under grant DMS 0099881.
The author is supported by KBN Grant 1 P03A 013 26.