Weighted Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces
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Weighted Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces Haibo Lin1 · Shengchen Mao1 Received: 26 July 2019 / Accepted: 3 November 2019 / © Springer Nature B.V. 2019
Abstract In this paper, the authors establish the weighted norm inequalities associated with the local Muckenhoupt weights for the local fractional integrals on Gaussian measure spaces. More precisely, the authors first obtain the weighted boundedness of local fractional integrals of order β from Lp (ωp ) to Lq (ωq ) for p ∈ (1, ∞) and from Lp (ωp ) to Lq,∞ (ωq ) for p = 1 under the condition of ω ∈ Ap,q,a , where 1/q = 1/p − β, and then obtain the weighted boundedness of the local fractional integrals, local fractional maximal operators and local Hardy-Littlewood maximal operators on the Morrey-type spaces over Gaussian measure spaces. Moreover, the method of proving the weighted weak type endpoint estimates of local fractional integrals is new. Keywords Local fractional integral · Local fractional maximal operator · Gaussian measure space · Morrey-type space Mathematics Subject Classification (2010) Primary 42B35 · Secondary 42B20, 42B25
1 Introduction Weighted norm inequalities have been widely used in various practical problems, for example, the study of boundary value problems for Laplace’s equation on Lipschitz domains, extrapolation theory, vector-valued inequalities, and estimates for certain classes of nonlinear partial differential equations; see, for instance, Grafakos [10]. In 1972, Muckenhoupt [25] introduced the Ap weights, which characterized the boundedness of Hardy-Littlewoood maximal operators on weighted Lebesgue spaces. In 1974, Muckenhoupt and Wheeden [26] introduced Ap,q weights and obtained the weighted boundedness of fractional integrals and This work is supported by the National Training Program of Innovation (Grant No. 201910019171). Shengchen Mao
[email protected] Haibo Lin [email protected] 1
College of Science, China Agricultural University, Beijing, 100083, People’s Republic of China
H. Lin, S. Mao
fractional maximal operators. After that, Welland [36] gave a simpler proof of the weighted boundedness of the fractional integrals. On the other hand, the Gaussian measure space (Rd , | · |, γ ) is the Euclidean space Rd setting with the Euclidean distance | · | and the Gaussian measure γ , where dγ (x) := π −d/2 e−|x| dx, 2
∀x ∈ Rd .
It is well known that γ is a natural measure associated with the differential operator 1 L0 (f )(x) := − f (x) + x · ∇f (x), ∀f ∈ Cc∞ (Rd ), 2 which is called the Ornstein-Uhlenbeck operator, and its closure L is self-adjoint on L2 (γ ); see, for instance, Sj¨ogren [32]. It should be remarked that, on Gaussian measure spaces, the operator behaves like the Laplacian, moreover, L is the infinitesimal generator of the Ornstein-Uhlenbeck semigroup. The Gaussian measure space is not a homogeneous type space in the sense of Coifman and Weiss [5], since there isn’t a positive constant C such that γ (B) Cγ (B/2) for any open ball B ⊂ Rd . Moreover, kernels of singul
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