Interpolation orbits in couples of Lebesgue spaces (Q2571492)

The present paper deals with interpolation for linear operators acting from an arbitrary couple \(\{L_{p_0}(U_0),L_{p_1}(U_1)\}\) of weighted \(L_p\)-spaces to another such couple \(\{L_{q_0}(V_0),L_{q_1}(V_1)\}\), where \(1\leq p_0,p_1,q_0,q_1\leq\infty\). The paper describes the orbits of an arbitrary element \(a\in L_{p_0}(U_0)+L_{p_1}(U_1)\). The main result says that \(\text{ Orb}(a,\{L_{p_0}(U_0),L_{p_1}(U_1)\}\to\{L_{q_0}(V_0), L_{q_1}(V_1)\})\) coincides with the space \(\phi(L_{q_0}(V_0),L_{q_1}(V_1))\), where \(\phi(X_0,X_1)_{r_0,r_1}\) is an interpolation functor, generalizing the Lions--Peetre method of means to the case of functional parameters \(\phi\). Actually, in the paper results announced in [\textit{V.~I.\ Ovchinnikov}, C.~R.\ Math.\ Acad.\ Sci.\ Paris 334, No.~10, 881--884 (2002; Zbl 1027.46091)] are proved.