The Approximation Functional and Interpolation with Perturbed Continuity Original Research Article

The continuity conditions at the endpoints of interpolation theorems, ‖Ta‖Bj⩽Mj ‖a‖Aj for j=0, 1, can be written with the help of the approximation functional: ‖E(t, Ta; B1, B0)‖L∞⩽M0 ‖a‖A0 and ‖E(t, Ta; B0, B1)‖L∞⩽M1 ‖a‖A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L∞ norms can be replaced by BMO(R+) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition f∈L0, i.e., f∗∈L∞, where f∗(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f∗∈BMO(R+).