On a theorem of Littlewood

In 1927 Littlewood constructed a bounded holomorphic function on the unit disc, having no tangential boundary limits almost everywhere. This theorem was the complement of a positive theorem of Fatou (1906), establishing almost everywhere non-tangential convergence of bounded holomorphic functions. There are several generalizations of Littlewood's theorem whose proofs are based on the specific properties of holomorphic functions. Applying real variable methods, we extend these theorems to general convolution operators.