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

Ebner Pineda; Luz Rodriguez; Wilfredo Urbina; Ebner Pineda; Luz Rodriguez; Wilfredo Urbina

doi:10.3934/math.20231388

AIMS Mathematics

2023, Volume 8, Issue 11: 27128-27150. doi: 10.3934/math.20231388

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

1.
Departamento de Matemáticas, Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL), Campus Gustavo Galindo km. 30.5 Vía Perimetral, Guayaquil EC090112, Ecuador
2.
Department of Mathematics and Actuarial Sciences, Roosevelt University, Chicago, IL, 60605, USA

Received: 28 April 2023 Revised: 29 August 2023 Accepted: 07 September 2023 Published: 25 September 2023
MSC : Primary 42B25, 42B35; Secondary 46E30, 47G10

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.
- Ornstein-Uhlenbeck,
- variable exponent,
- Besov-Lipschitz,
- Triebel-Lizorkin,
- Gaussian measure
Citation: Ebner Pineda, Luz Rodriguez, Wilfredo Urbina. Variable exponent Besov-Lipschitz and Triebel-Lizorkin spaces for the Gaussian measure[J]. AIMS Mathematics, 2023, 8(11): 27128-27150. doi: 10.3934/math.20231388

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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.

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