Variable exponent Besov-Lipschitz and Triebel-Lizorkin spaces for the Gaussian measure

Abstract

In this paper, we introduce variable Gaussian Besov-Lipschitz $ B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d}) $ and Triebel-Lizorkin spaces $ F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d}), $ i.e., Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces with variable exponents $ p(\cdot) $ and $ q(\cdot) $, under certain regularity conditions on the functions $ p(\cdot) $ and $ q(\cdot) $. The condition on $ p(\cdot) $ is associated with the Gaussian measure and was introduced in [3]. Trivially, they include the Gaussian Besov-Lipschitz $ B_{p, q}^{\alpha}(\gamma_{d}) $ and Triebel-Lizorkin spaces $ F_{p, q}^{\alpha}(\gamma_{d}) $ for $ p, q $ constants, which were introduced and studied in [10]. We consider some inclusion relations of those spaces and finally prove some interpolation results for them.

Keywords

• ornstein-uhlenbeck
• variable exponent
• besov-lipschitz
• triebel-lizorkin
• gaussian measure