Interpolation orbits in the Lebesgue spaces

This paper is devoted to description of interpolation orbits with respect to linear operators mapping an arbitrary couple of L p spaces with weights {L p 0 (U 0), L p 1 (U 1)} into an arbitrary couple {L q 0 (V 0), L q 1 (V 1)}, where 1 ≤ p 0 , p 1 , q 0 , q 1 ≤ ∞. By L p (U) we denote the space of measurable functions f on a measure space M such that f U ∈ L p with the norm f Lp(U) = f U Lp. }) is a Banach space of y ∈ Y 0 + Y 1 such that y = T a, where T is a linear operator mapping the couple {X 0 , X 1 } into the couple {Y 0 , Y 1 }. The norm is defined by y Orb = inf T max(T X 0 →Y 0 , T X 1 →Y 1), where infimum is taken over all representations of y in the form y = T a. This space is called an interpolation orbit of the element a. We are going to describe the spaces Fundamental results on description of these spaces in separate cases are well known since 1965. The key role was played by the J.Peetre K-functional. x=x 0 +x 1 sx 0 X 0 + tx 1 X 1 where infimum is taken over all representations of x as a sum of x 0 ∈ X 0 and x 1 ∈ X 1. The function K(s, t) is concave and it is uniquely defined by the function (U 1)}), for any a ∈ L p 0 (U 0) + L p 1 (U 1). First results were obtained by B.S.Mitiagin and A.P.Calderon in the case p 0 = q 0 = 1 and p 1 = q 1 = ∞. The final steps in the description were done by G. then there exists a linear operator T : {L p 0 (U 0), L p 1 (U 1)} → {L p 0 (V 0), L p 1 (V 1)} such that y = T a.