## Tohoku Mathematical Journal 2000 March SECOND SERIES VOL. 52, NO. 1

 Tohoku Math. J. 52 (2000), 31-46

Title MODULAR INEQUALITIES FOR THE CALDERÓN OPERATOR

Author María J. Carro and Hans Heinig

(Received June 1, 1998, revised January 5, 1999)
Abstract. If $P,Q:[0,\infty)\to$ are increasing functions and $T$ is the Calderon operator defined on positive or decreasing functions, then optimal modular inequalities $\int P(Tf)\leq C\int Q(f)$ are proved. If $P=Q$, the condition on $P$ is both necessary and sufficient for the modular inequality. In addition, we establish general interpolation theorems for modular spaces.

1991 Mathematics Subject Classification. Primary 46M35; Secondary 46E30.