Under some assumptions on the pair $(A_0,B_0)$, we study equivalence between interpolation properties of linear operators and monotonicity conditions for a pair (Y,Z) of rearrangement invariant quasi-Banach spaces when the extreme spaces of the interpolation are $L^{\infty }$. Weak and restricted weak intermediate spaces fall within our context. Applications to classical Lorentz and Lorentz-Orlicz spaces are given.

If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K(t,f;L^p\left(C\right),L^{\infty }\left(C\right))$ to quasicontinuous functions f ∈ QC is equivalent to $K(t,f;L^p\left(C\right)\cap QC,L^{\infty }\left(C\right)\cap QC)$. We apply this result to identify the interpolation space ${(L^{p₀,q₀}\left(C\right)\cap QC,L^{p₁,q₁}\left(C\right)\cap QC)}_{\theta ,q}$.

Certain operator ideals are used to study interpolation of operators between spaces generated by the real method. Using orbital equivalence a new reiteration formula is proved for certain real interpolation spaces generated by ordered pairs of Banach lattices of the form $(X,L_{\infty }\left(w\right))$. As an application we extend Ovchinnikov’s interpolation theorem from the context of classical Lions-Peetre spaces to a larger class of real interpolation spaces. A description of certain abstract J-method spaces is also presented....

The behavior of the essential spectrum and the essential norm under (complex/real) interpolation is investigated. We extend an example of Albrecht and Müller for the spectrum by showing that in complex interpolation the essential spectrum ${\sigma }_e\left(S_{\left[\theta \right]}\right)$ of an interpolated operator is also in general a discontinuous map of the parameter θ. We discuss the logarithmic convexity (up to a multiplicative constant) of the essential norm under real interpolation, and show that this holds provided certain compact approximation...

We study the behavior of the ball measure of non-compactness under several interpolation methods. First we deal with methods that interpolate couples of spaces, and then we proceed to extend the results to methods that interpolate finite families of spaces. We will need an approximation hypothesis on the target family of spaces.

We investigate the behaviour of the measure of non-compactness of an operator under real interpolation. Our results refer to general Banach couples. An application to the essential spectral radius of interpolated operators is also given.

We study a problem of interpolating a linear operator which is bounded on some family of characteristic functions. A new example is given of a Banach couple of function spaces for which such interpolation is possible. This couple is of the form $\overline{\Phi }=(B,L^{\infty })$ where B is an arbitrary Banach lattice of measurable functions on a σ-finite nonatomic measure space (Ω,Σ,μ). We also give an equivalent expression for the norm of a function ⨍ in the real interpolation space ${(B,L^{\infty })}_{\theta ,p}$ in terms of the characteristic functions of...

The aim of the paper is to get an estimation of the error of the general interpolation rule for functions which are real valued on the interval $[-a,a]$, $a\in (0,1)$, have a holomorphic extension on the unit circle and are quadratic integrable on the boundary of it. The obtained estimate does not depend on the derivatives of the function to be interpolated. The optimal interpolation formula with mutually different nodes is constructed and an error estimate as well as the rate of convergence are obtained. The general...

A scale of function spaces is considered which proved to be of considerable importance in analysis. Interpolation properties of these spaces are studied by means of the real interpolation method. The main result consists in demonstrating that this scale is interpolated in a way different from that for Lp spaces, namely, the interpolation space is not from this scale.

We introduce various classes of interpolation sets for Fréchet measures-the measure-theoretic analogues of bounded multilinear forms on products of C(K) spaces.

We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces ${ℒ}^s\left(ℝⁿ\right)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation ${div\left|\nabla u\right|}^{p-2\nabla }u=div$. In this example the p-harmonic transform is essentially inverse to $div\left(\right|\nabla {|}^{p-2}\nabla )$. To every vector field $\in {ℒ}^q(ℝⁿ,ℝⁿ)$ our operator ${\mathscr{H}}_p$ assigns the gradient of the solution, ${\mathscr{H}}_p=\nabla u\in {ℒ}^p(ℝⁿ,ℝⁿ)$. The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our arguments...