Interpolation Orbits in Couples of Lebesgue Spaces

The paper deals with interpolation orbits for linear operators acting from an arbitrary couple $\{L_{p_0}(U_0),L_{p_1}(U_1)\}$ of weighted $L_p$ spaces into an arbitrary couple $\{L_{q_0}(V_0),L_{q_1}(V_1)\}$ of such spaces, where $1\le p_0,p_1,q_0, q_1\le\infty$. Here $L_p(U)$ is the space of measurable functions $f$ on a measure space such that $fU\in L_p$, equipped with the norm $\|f\|_{L_p(U)}=\|fU\|_{L_p}$. The paper describes the orbits of arbitrary elements $a\in L_{p_0}(U_0)+L_{p_1}(U_1)$. It contains proofs of the results announced in C. R. Acad. Sci. Paris, Ser. I, 334, 881–884 (2002).