Gyro- L p spaces

Source abstract

Abstract In this paper we introduce a gyro-analogue of L p {L^{p}} spaces; we achieve this goal by constructing normed gyrolinear structures on the set of functions from an arbitrary measure space to the Möbius gyrovector space ℳ s {\mathcal{M}_{s}} and the Einstein gyrovector space ℰ s {\mathcal{E}_{s}} . This gives rise to two major classes of gyro- L p {L^{p}} spaces: Möbius s - L p {L^{p}} gyrospaces and Einstein s - L p {L^{p}} gyrospaces. These two spaces will form a normed gyrolinear space with respect to the ∥ ⋅ ∥ p ℳ s {\|\cdot\|_{p}^{\mathcal{M}_{s}}} gyronorm and the ∥ ⋅ ∥ p ℰ s {\|\cdot\|_{p}^{\mathcal{E}_{s}}} gyronorm, respectively. We show that the Einstein L p {L^{p}} -gyrospaces and Möbius L p {L^{p}} -gyrospaces are isomorphic, and the isomorphism between them also preserves the L p {L^{p}} gyronorm, thus giving rise to a gyroisometry. Furthermore, we show that the Möbius and Einstein L p {L^{p}} gyronorms induce a complete metric space structure on the Möbius L p {L^{p}} gyrospace and the Einstein L p {L^{p}} gyrospace, respectively.