UDc 517.982 the oPtimal emBeDDing for the calDeron tyPe sPaces anD the J-methoD sPaces a. gogatishvili* , V. i. ovchinnikov * Mathematical Institute, Czech Academy of Sciences Voronezh State University Calderon spaces L( , )F E in terms of the J-method interpolation spaces is found. we show that the corresponding lorentz space LE AMS classification 46M35, 46E30, 46B70. <...> Key words: embedding theorems, optimal spaces for embedding, interpolation spaces, real method spaces introDUction we consider embeddings of the Calderon spaces L( , )E F to rearrangement invariant spaces. <...> The Calderon spaces are defined with the help of the best approximation e ft E functions of exponential type of degree t n1/ t E F ( ) of f EŒ by entire in each variable in the norm of rearrangement invariant space E . <...> The space L( , )F E , where F is a functional lattice on (0, )• , consists of f EŒ that e f( ) Œ with the corresponding norm or such quasi-norm. <...> These spaces are intimately connected with the Besov spaces and their generalizations. <...> Thus embeddings L( , )E F XГ are studied along with the study of the embeddings of smooth function spaces (e.g., see [3, 5, 6]). <...> M.Goldman and R.Kerman in [7] found the rearrangement invariant space X0 which envelope the space L( , )E F . <...> In the present paper we give a new description of X0 terms of interpolation spaces. we find that X0 in is described as a J -method interpolation space with a concrete parameter between some lorentz space and L• . , thus we clarify the position of this space in the family of rearrangement invariant spaces. <...> Some conditions added to the conditions of [7] enable us to give a very transparent description of the optimal space X0 Basic Definitions anD notation Everywhere below we use notation from [7]. <...> Denote by f * the decreasing rearrangement of a © Gogatishvili A., Ovchinnikov V. I., 2006 * The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research plan No. AV0Z 0 90503. <...> A Banach lattice is a space E of measurable functions with a monotone norm, i.e. f g,£ Œ fi Œ , f g g <...>