Estimates of solutions of the $H^p$ and BMOA corona problem

We prove new sharper estimates of solutions to the $H^p$ -corona problem in strictly pseudoconvex domains; in particular we show that the constant is independent of the number of generators. We also obtain sharper estimates for solutions to the BMOA corona problem. The proofs also lead to new results about the Taylor spectrum of analytic Toeplitz operators on $H^p$ and BMOA.