The approximation functional and interpolation with perturbed continuity

The continuity conditions at the endpoints of interpolation theorems, \\Ta\\(B1)less than or equal to M-j \\a\\(Aj) for j = 0, 1 can be written with the help of the approximation functional: \\E( t, Td; B-1; B-0)\\ (L infinity) less than or equal to M-0 \\ a\\ (A0) and \\E( i, Ta; B-0, B-1)\\ (L infinity) less than or equal to M-1 \\a\\(A1). As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L-infinity norms can be replaced by BMO(R+.) norms. This leads to a strong version of the Srein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition f epsilon L-0, i.e., f(*) epsilon L-infinity, where f(*)(gamma) = mu{\f\ > gamma} is the distribution function of f, can be replaced in interpolation with L.(p, q) spaces by the weaker f(*) epsilon BMO(R+). (C) 2001 Academic Press.