The Besov Subspace Consisting of Most Non-Smooth Functions

Under the assumptions that Delta(f, h)(t) = vertical bar f(t + h) - f(t)vertical bar, X is a symmetric space of functions in [0, 1], alpha is an element of (0, 1) and p is an element of [1,8) are any fixed number, by the triple (X, alpha, p) a Besov type space Lambda(alpha)(X,p) is constructed, where the norm is given by the equality parallel to f vertical bar Lambda(alpha)(X,p)parallel to = ((i=1)Sigma(infinity)(2(alpha i)parallel to Delta(f;2(-1))(.)vertical bar X parallel to)(p))(1/p). For any alpha(0) is an element of (0,1), it is shown that there exists an infinite-dimensional, closed subspace of Lambda(alpha)(X,p), such that any non-identically zero function does not belong to the subspace Lambda(alpha)(X,p) with alpha > alpha(0).