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 $(0\<p\le \infty )$. 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...