Inclusions and the approximate identities of the generalized grand Lebesgue spaces

Abstract

Let (Omega, Sigma, mu) and (Omega, Sigma, upsilon) be two finite measure spaces and let L-p(),theta )(mu) and L-q),L-theta (upsilon) be two generalized grand Lebesgue spaces [9,10] , where 1 < p, q < infinity and theta >= 0. In Section 2 we discuss the inclusion properties of these spaces and investigate under what conditions L-p),L-theta (mu) subset of L-q),L-theta (upsilon) for two different measures mu and upsilon. Let Omega be a bounded subset of R-n. We know that the Lebesgue space L-p (mu) admits an approximate identity, bounded in L-1 (mu) , [5, 8] . In Section 3 we investigate the approximate identities of L-p),L-theta (mu) and show that it does not admit such an approximate identity. Later we discuss aproximate identities of the space [L-p](p)),(theta) , the closure of C-c(infinity) (Omega) in L-p),L-theta (mu), where C-c(infinity) (Omega) denotes the space of infinitely differentiable complex-valued functions with compact support on R-n.