Interpolation orbits in couples of Lebesgue spaces

We consider linear operators $T$ mapping a couple of weighted $L_p$ spaces $\{L_{p_0}(U_0),L_{p_1}(U_1)\}$ into $\{L_{q_0}(V_0),L_{q_1}(V_1)\}$ for any $1\le p_0$, $p_1$, $q_0$, $q_1\le \mathrm{\infty }$, and describe the interpolation orbit of any $a\in L_{p_0}(U_0)+L_{p_1}(U_1)$ that is we describe a space of all $\{Ta\}$, where $T$ runs over all linear bounded mappings from ${L_{p_0}(U_0), L_{p_1}(U_1)}$into${L_{q_0}(V_0),L_{q_1}(V_1)}$. We present in this paper the proofs of results which were announced in V.I.Ovchinnikov, C. R. Acad. Sci. Paris Ser. I 334 (2002) 881–884. We show that interpolation orbit is obtained by the Lions–Peetre method of means with functional parameter as well as by the K-method with a weighted Orlicz space as a parameter.