Abstract and Concrete Logarithmic Interpolation Spaces

Abstract

A procedure is given to reduce the interpolation spaces on an ordered pair generated by the function parameter t θ (1 + |log t|) −b to the classical real interpolation spaces. Applications are given for Lorentz-Zygmund function spaces, Besov spaces of generalized smoothness and Lorentz-Zygmund operator spaces.

On an extreme class of real interpolation spaces

“…where 0 < θ = 1 p < 1, 1 q ∞ and b ∈ R. Using this last formula one can describe LorentzZygmund spaces in terms of Lorentz spaces (see [12,30,10]). …”

“…They are also limit cases of logarithmic interpolation spaces in the terminology of [12]. The following characterization can be established by using the same arguments as in [12,Theorem 1].…”

Extrapolation estimates for entropy numbers

“…The extrapolation spaces that we are going to use are based on ideas of [19,17,18,20,8,7]. Definition 2.1.…”

Observation of Volterra systems with scalar kernels

“…where A = (0, α), see also Cobos, Frenandez-Cabrera and Triebel [7] for logarithmic type interpolation functors. In some situations, the condition of sectoriality of angle < π / 2 in the above corollary may be inconvenient.…”