Interpolation of weighted and vector-valued Hardy spaces

Abstract

Real and complex interpolation methods, when applied to the couple ( H p 0 ( E 0 ; w 0 ) , H p 1 ( E 1 ; w 1 ) ) ({H^{{p_0}}}({E_0};{w_0}),{H^{{p_1}}}({E_1};{w_1})) , give what is expected if E 0 {E_0} and E 1 {E_1} are quasi-Banach lattices of measurable functions satisfying certain mild conditions and if log ⁡ ( w 0 1 / p 0 w 1 − 1 / p 1 ) ∈ BMO ( w 0 , w 1 \log (w_0^{1/{p_0}}w_1^{ - 1/{p_1}}) \in {\text {BMO}}\,({w_0},{w_1} being weights on the unit circle). The last condition is in fact necessary. (It is expected, of course, that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple ( L p 0 ( E 0 ; w 0 ) , L p 1 ( E 1 ; w 1 ) ) . ) ({L^{{p_0}}}({E_0};{w_0}),{L^{{p_1}}}({E_1};{w_1})).)