On the optimal embedding of Calderón spaces and of generalized Besov spaces

In this paper embeddings of spaces of Calderón type Lambda(E,F) into rearrangement-invariant spaces X on Bbb{R}^n are studied. More precisely, let E=E(Bbb{R}^n) be some rearrangement-invariant space. Then Lambda(E,F) contains those fin E such that for its best approximation in the norm of E with respect to entire functions of exponential type t^{1/n}, multline e_t(f)_E = infleft{ |f-q|_E colon qin E, roman{supp} scr{F}q subset (-t^{1/n},t^{1/n})^nright}, quad t>0,endmultline we have e_t(f)_E in F=F(Bbb{R}_+) = L(varrho)cap L_infty, the space of measurable functions on Bbb{R}_+ that are essentially bounded. Here scr{F} stands for the Fourier transform on Bbb{R}^n, as usual. Note that this setting contains as special cases not only (classical) Besov spaces and spaces of generalised smoothness, but also some structural and constructive spaces of Calderón type considered by Goldʹman before. Main results of this paper concentrate on necessary and sufficient conditions for the embedding Lambda(E,F) subset X, where X=X(Bbb{R}^n) stands for some (arbitrary) rearrangement-invariant space. The given criteria rely on the embedding Ecap L_infty subset X as well as the boundedness of some Hardy operator of type H[g](t) = int_0^{1/t} g(tau) psi_E(tau) dtau, quad gin Omega_F, quad t>0, acting from the (quasi-) normed cone Omega_F = Omegacap F into X(0,1), where Omega is the cone of bounded, monotonically decreasing, non-negative functions on Bbb{R}_+, Omega = left{ g colon 0leq g(t) downarrow, lim_{tto +0} g(t)