Interpolation between weighted Hardy spaces

Abstract

We prove that H p ( w 0 1 − s w 1 s ) {H^p}(w_0^{1 - s}w_1^s) is an interpolation space of exponent s s between H p ( w 0 ) {H^p}({w_0}) and H p ( w 1 ) {H^p}({w_1}) if and only if log ⁡ ( w 1 / w 0 ) \log ({w_1}/{w_0}) is in BMO. If log ⁡ ( w 1 / w 0 ) \log ({w_1}/{w_0}) fails to be in BMO, H p ( w 0 1 − s w 1 s ) {H^p}(w_0^{1 - s}w_1^s) can still be an interpolation space, provided the range of w w has sufficiently large gaps.