Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators Ta with locally integrable symbols on Bergman spaces Ap(D), 1 < p < 1. Our main result gives a sucient condition for the bounded- ness of Ta in terms of some "averages" (related to hyperbolic disks) of its symbol. If the averages satisfy an o-type condition on the boundary of D, we show that the corresponding Toeplitz opera- tor is compact on Ap. Both conditions coincide with the known necessary conditions in the case nonnegative symbols and p = 2. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on Ap provided that the averages are bounded away from zero, and derive an index formula for these operators.