Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

By The Anh Bui, Xuan Thinh Duong, Fu Ken Ly. Transactions of the American Mathematical Society, 2016

89 citations recorded by the source corpus.

Let X X be a space of homogeneous type and let L \mathfrak {L} be a nonnegative self-adjoint operator on L 2 ( X ) L^2(X) enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove (local) nontangential and radial maximal function characterizations for the local Hardy spaces associated to L \mathfrak {L} . This gives the maximal function characterization for local Hardy spaces in the sense of Coifman and Weiss provided that L \mathfrak {L} satisfies certain extra conditions. Secondly we introduce local Hardy spaces associated with a critical function ρ \rho which are motivated by the theory of Hardy spaces related to Schrödinger operators and of which include the local Hardy spaces of Coifman and Weiss as a special case. We then prove that these local Hardy spaces can be characterized by (local) nontangential and radial maximal functions related to L \mathfrak {L} and ρ \rho , and by global maximal functions associated to ‘perturbations’ of L \mathfrak {L} . We apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Schrödinger operators on manifolds to Schrödinger operators on connected and simply connected nilpotent Lie groups.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f2494c472dc24ac709309d0b046deeda7e2a261c

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. The Anh Bui
4. Xianjie Yan
5. Xuan Thinh Duong
6. H. Bui
7. Ziyi He
8. Fu Ken Ly

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. The Anh Bui
4. Xuan Thinh Duong
5. Xianjie Yan
6. H. Bui
7. Ziyi He
8. Fu Ken Ly

Citations in the corpus

• Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces (2021, background): “We refer the reader to [8, 9, 12, 22, 28, 31, 34, 45, 49, 72] for more studies on the real-variable theory of function spaces over and to [18, 46, 51, 73] for their applications in the boundedness of operators; particularly, we refer the reader to [6, 7, 10] for some recent progress on the real-var”
• WEIGHTED BESOV AND TRIEBEL–LIZORKIN SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS (2020, background): “For the items (b) and (c), we refer to [21].”
• Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications (2018, background): “ : X∞ k=1 λ4 k,ν|hf,ψ ν ki| 2 <∞}. This operator is called the Bessel operator on ((0,1),dx). In order to consider the maximal function characterization for the Hardy spaces associated to L, as in [4] we consider the intervals: (47) Jj = ( (1 −2−j,1 −2−j−1], j≥ 1 (2j−1,2j], j≤ −1. which are depic”