Stein interpolation for the real interpolation method

Abstract

We prove a complex formulation of the real interpolation method, showing that the real and complex interpolation methods are not inherently real or complex. Using this complex formulation, we prove abstract Stein interpolation for the real interpolation method. We apply this theorem to interpolate weighted L p -spaces and the sectoriality of closed operators with the real interpolation method.

A discrete framework for the interpolation of Banach spaces

“…Using the complex formulation of the sequentially structured interpolation method, we will now prove abstract Stein interpolation for any two compatible couples of sequentially structured Banach spaces (X 0 , X 1 ) and (Y 0 , Y 1 ). In particular we obtain abstract Stein interpolation for the real interpolation method, which we already treated in a continuous setting in [LL21a]. Note that abstract Stein interpolation for the abstract framework in [CKMR02] was posed as an open problem, see [Kal16,p.662].…”

“…In [LL21b] we will develop such an abstract continuous framework. In [LL21a] we already highlighted the specific case of real interpolation. Now let us return to the question in (4.3), which boils down to the question whether (X 0 , X 1 ) (c) θ defines a Banach space.…”

“…Alternatively one can use a continuous framework for interpolation, which does not require such a periodicity assumption. For real interpolation this was done in [LL21a] and we will develop this abstractly in [LL21b]. (iv) The implicit constant in the conclusion of Theorem 5.5 is a consequence of both our discrete framework and the approximation in Lemma 3.8.…”

“…The power of Theorem 5.5 is of course that it also works for other interpolation methods. For example, using abstract Stein interpolation for the real interpolation method we interpolated weighted L p -spaces and analytic semigroups in [LL21a]. In Lemma 6.13 we will use abstract Stein interpolation for general sequence structured interpolation to interpolate weighted sequence spaces S j (a j ) for j = 0, 1 to deduce a reiteration theorem.…”

“…This theorem is well-known for the complex interpolation method and was proven for the γ-interpolation method in [SW06]. For the specific case of real interpolation, we already used similar ideas in a continuous setting in [LL21a]. We remark that for the interpolation framework developed in [CKMR02], abstract Stein interpolation was phrased as an open problem [Kal16,p.662].…”