The Fredholm properties of Toeplitz operators on the Bergman space A² have been well-known for continuous symbols since the 1970s. We investigate the case p=1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on A^p that arise when we no longer have 1<p<∞; in particular bounded Toeplitz operators on A¹ were characterized completely very recently but only for bounded symbols. We also consider compactness of Hankel operators on A¹.

The first author was partially supported by the Väisälä foundation of the Finnish Academy of Science and Letters. The second author was supported by Academy of Finland Project 207048.