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We study the relation of autoreducibility and mitoticity for polylog-space many-one reductions and log-space many-one reductions. For polylog-space these notions coincide, while proving the same for log-space is out of reach. More precisely, we show the following results with respect to nontrivial sets and many-one reductions:1.polylog-space autoreducible @? polylog-space mitotic, 2.log-space mitotic @? log-space autoreducible @? (logn@?loglogn)-space mitotic, 3.relative to an oracle, log-space autoreducible @? log-space mitotic. The oracle is an infinite family of graphs whose construction combines arguments from Ramsey theory and Kolmogorov complexity.