Generalized Interpolation Methods

The origin of the theory of interpolation can be traced back to Marcinkiewicz and the Riesz-Thorin theorem, which states that if a linear function is continuous on L^p and L^q, then it is also continuous on L^r for r between p and q. Later, as it was shown that Sobolev spaces were constituted of functions that have a non-integer order of differentiability, various techniques were conceived to generate similar spaces. Among them were the interpolation methods, which have been generalized using a function parameter. Most of the times, we start from the K-method. Let A_0 and A_1 be two Banach spaces continuously embedded into a Hausdorff topological vector space so that A_0 ∩ A_1 and A_0 + A_1 are well-defined Banach spaces. One defines the K-functional by: K(t,a) := infₐ₌ₐ₀₊ₐ₁ { ∥a₀∥{A₀} + t ∥a₁∥{A₁} }, for t>0 and a ∈ A_0 + A_1. Given 0<θ<1 and q∈[1,∞], a belongs to the interpolation space if a∈ A _0 ​+ A_1 ​ and (2^{-jθ} K(2^j, a))_{j ∈ ℤ} ∈ ℓ^q. The generalized version is obtained by replacing the sequence (2^{-jθ})_{j∈Z} appearing in the expression above with a Boyd function. The J-method is defined in a similar way and one can show that both methods give rise to the same spaces. In this work, we show that the Boyd functions form a natural apparatus for studying function spaces and that interpolation methods with a function parameter provide an interesting tool in this context. For example, they lead to a definition of the Besov spaces of generalized smoothness based on the usual Sobolev spaces.