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AbstractTheorems 1 and 2 are known results concerning Lp?Lqestimates for certain operators wherein the point (1/p, 1/q) lies on the line of duality 1/p+ 1/q= 1. In Theorems 1? and 2? we show that with mild additional hypotheses it is possible to prove Lp-Lqestimates for indices (1/p, 1/q) off the line of duality. Applications to Bochner-Riesz means of negative order and uniform Sobolev inequalities are given.

Countable interpolation sets of type 1 are characterized as those countable compact sets which intersect each part at most once. A metric condition is given which characterizes those countable compact sets which are interpolation sets for logmodular function algebras.

We study the peak sets for the algebras of functions analytic in the unit disc D and satisfying a Lipschitz condition on .

We prove a unique continuation theorem for a class of differential operators containing the heat operator.

Representing measures are employed to give a short proof of the following theorem: A normed linear space is reflexive if its unit ball is weakly compact.