A VECTOR-VALUED H p -CORONA THEOREM ON THE POLYDISK TAVAN T. TRENT Abstract. For the corona problem on the bidisk, we find analytic solutions belonging to the Orlicz-type space exp(L 1 3 ). In addition, for 1 ≤ p< ∞, an H p (D 2 ) corona theorem is established. Similar techniques can be used for the polydisk. In this paper we give a solution, for general corona problem data on the bidisk, which, although not bounded or even in BMO, still belongs to a space better than  ∞ p=1 H p (D 2 ); namely the Orlicz-type space, exp(L 1 3 ) (for the bidisk case). Also, we establish the H p -corona theorem on the bidisk. For the general polydisk, similar methods can be applied. For the case of two functions in the input data, Chang [6] showed that solutions to the general corona problem for the bidisk can be found which belong to  ∞ p=1 H p (D 2 ). Again for two functions on the bidisk, Amar [1] and Cegrell [5] have found solutions to the general corona problem for the bidisk belonging to H ∞ -BMO. For a finite number of input functions, the ∂ -input data is more complicated and, for this case, first Varopoulos [25] and then Lin [18] found solutions to the gen- eral corona problem on the polydisk belonging to  ∞ p=1 H p (D n ). [See Chang and R. Fefferman [8] for a brief discussion of the difference (in- volving the Koszul complex) between two and a general finite number of input functions.] However, even in this case no relationship between the lower bound of the input data (denoted by ǫ) and the size of the solutions was obtained. An estimate will be given in this paper. For a finite number of input functions, Li [17] and, independently, Lin [18] implicitly solved the H p (D n ) corona theorem (1 ≤ p< ∞) based on the work of Lin. Again, for a finite number of input functions, Boo [3] gave an explicit solution to the H p (D n ) corona theorem (1 ≤ p< ∞), which was based on integral formulas. For the case p = 2, we Key words and phrases. corona theorem, polydisk. Partially supported by NSF Grant DMS-0100294. 1