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...

We prove that the basic facts of the real interpolation method remain true for couples of cones obtained by intersection of the cone of concave functions with rearrangement invariant spaces.

We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted $L_p$-spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability...

We study the measure of non-compactness of operators between abstract real interpolation spaces. We prove an estimate of this measure, depending on the fundamental function of the space. An application to the spectral theory of linear operators is presented.

Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is $T:A_{\left[\theta \right]}\to B_{\left[\theta \right]}$ for all 0 < θ < 1, where $A_{\left[\theta \right]}$ and $B_{\left[\theta \right]}$ are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.

We prove that the fractional BMO space on a one-dimensional manifold is an interpolation space between C and $C^1$. We also prove that $BM{O}^1$ is an interpolation space between C and $C^2$. The proof is based on some nonclassical interpolation constructions. The results obtained cannot be transferred to spaces of functions defined on manifolds of higher dimension. The interpolation description of fractional BMO spaces is used at the end of the paper for the proof of the boundedness of commutators of the Hilbert...

Let $H_p$ denote the usual Hardy space of analytic functions on the unit disc $(0\&lt;p\le \infty )$. We prove that for every function $f\in H_1$ there exists a linear operator $T$ defined on $L_1\left(\mathbf{T}\right)$ which is simultaneously bounded from $L_1\left(\mathbf{T}\right)$ to $H_1$ and from $L_{\infty }\left(\mathbf{T}\right)$ to $H_{\infty }$ such that $T\left(f\right)=f$. Consequently, we get the following results $(1\le p_0,p_1\le \infty )$:1) $(H_{p_0},H_{p_1})$ is a Calderon-Mitjagin couple;2) for any interpolation functor $F$, we have $F(H_{p_0},H_{p_1})=H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$, where$H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$ denotes the closed subspace of $F\left(L_{p_0}\right(\mathbf{T}),L_{p_1}(\mathbf{T}\left)\right)$ of all functions whose Fourier coefficients vanish on negative integers.These results also extend to Hardy...

We establish interpolation formulæ for operator spaces that are components of a given quasi-normed operator ideal. Sometimes we assume that one of the couples involved is quasi-linearizable, some other times we assume injectivity or surjectivity in the ideal. We also show the necessity of these suppositions.