An Interpolation Theorem with Perturbed Continuity

Abstract

Let (A0, A1) and (B0, B1) be two interpolation couples and let T:(A0, A1)↦ (B0, B1) be a K-quasilinear operator. The boundedness of the operator from A0 to B0 implies K(t, Ta; B0, B1)⩽M0‖a‖A0 and the boundedness of the operator from A1 to B1 implies K(t, Ta;B1, B0)⩽M1‖a‖A1, a∈A0∩A1. We consider perturbations of these two inequalities in the formK( γt, Ta; B 0, B 1)⩽M0‖a‖ A0+ε 0K (t, Ta;B 0, B 1) andK(γt, Ta;B 1, B 0)⩽M1‖a‖ A1+ε1K (t, Ta;B1, B0). We prove that similar to the classical case for γ>1 and 0⩽εj⩽1 we get for all 0<θ<1 and for all 0<q⩽∞‖Ta‖(B0, B1)θ , q⩽C(M0, M1, θ, q)‖a‖(A0, A1)θ, q. If we take B0=L1(Q), B1=L∞(Q), γ=2, and εj=1, where Q is a cube in Rn, we get a theorem of Bennett, DeVore, and Sharpley. We prove that if εj>1 we continue to get an interpolation theorem, but the interpolation holds for (log+ε0)/(logγ)<θ<1−(log+ε1)/(logγ). This is the first instance of an interpolation theorem which holds for a subinterval of 0<θ<1. Bennett, DeVore and Sharpley identified a “weak L∞” class as the rearrangement invariant span of BMO(Q). This prompts the natural question of the existence of abstract “weak type” classes near the endpoints of interpolation scales. As we note below, there have been previous attempts to develop the theory of such classes. The construction was, however, too rigid, and necessitated the precise identification of the K -functional for the interpolation couple. We define these classes here in a way which allows for the identification K-functionals up to multiplicative equivalence. This opens the door to applications of the theory to most interpolation couples, leading to stronger interpolation theorems even for some well known spaces.