Hartmanis conjectured that there exist no sparse complete sets for P under logspace many-one reductions. In this paper, in support of the conjecture, at as shown that if P has sparse hard sets under logspace many-one reductions, then P E DSPACE[log2n]. The result follows f rom a more general statement: i f P has 2P01Y'0g sparse hard sets under poly-logarithmic space-computable many-one reductions, then P C DSPACE[poZylog] .