Iterated Weak and Weak Mixed-Norm Spaces with Applications to Geometric Inequalities
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Iterated Weak and Weak Mixed-Norm Spaces with Applications to Geometric Inequalities Ting Chen1 · Wenchang Sun1 Received: 22 December 2018 © Mathematica Josephina, Inc. 2019
Abstract In this paper, we consider two types of weak norms, the weak mixed-norm and the iterated weak norm, in Lebesgue spaces with mixed norms. We study properties of two weak norms and present their relationship. Even for the ordinary Lebesgue spaces, the two weak norms are not equivalent and any one of them can not control the other one. We give some convergence and completeness results for the two weak norms, respectively. We study the convergence in the truncated norm, which is a substitution of the convergence in measure for mixed-norm Lebesgue spaces. And we give a characterization of the convergence in the truncated norm. We show that Hölder’s inequality is not always true on weak mixed-norm Lebesgue spaces and we give a complete characterization of indices which admit Hölder’s inequality. As applications, we establish some geometric inequalities related to fractional integrals in weak mixednorm spaces and in iterated weak spaces, respectively, which essentially generalize the Hardy–Littlewood–Sobolev inequality. Keywords Iterated weak norms · Weak mixed norms · Geometric inequalities Mathematics Subject Classification 42B35 · 26D15
1 Introduction For p = ( p1 , p2 ) and a measurable function f (x1 , x2 ) defined on Rn × Rm , where p1 , p2 are positive numbers and n, m are positive integers, we define the L p norm of f by
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Wenchang Sun [email protected] Ting Chen [email protected]
1
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
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T. Chen, W. Sun
f L p := f L xp1 1
p
L x22
.
The Lebesgue space L p (Rn × Rm ) with mixed norms consists of all measurable functions f for which f L p < ∞. We also write L p as L p2 (L p1 ). Lebesgue spaces with mixed norms were first studied systematically by Benedek and Panzone [5], where many fundamental properties were proved. In particular, they showed that such spaces possess similar properties as usual Lebesgue spaces. See also related works by Hörmander [24], Benedek et al. [4], Rubio de Francia et al. [35], and Fernandez [17]. Recently, many works have been done for mixed-norm spaces. For example, Stefanov and Torres [36] proved the boundedness of Calderón–Zygmund operators on mixed-norm Lebesgue spaces. Kurtz [32] proved that some classical operators, which include the strong maximal function, the double Hilbert transform and singular integral operators, are bounded on weighted Lebesgue spaces with mixed norms. Johnsen and Sickel studied the trace problem for Triebel–Lizorkin spaces with mixed norms [30] and gave a direct proof of Sobolev embeddings for quasi-homogeneous Triebel– Lizorkin spaces with mixed norms [31]. Johnsen et al. [27–29], Georgiadis and Nielsen [19] and Georgiadis et al. [18] gave various properties of anisotropic Triebel–Lizorkin spaces with mixed norms. Torres and Ward [39] gave Calderón’s reprodu
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