Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains

By D. Deng, Liang Song, Chaoqiang Tan, Lixin Yan. 2005

373 citations recorded by the source corpus.

Let L be the infinitesimal generator of an analytic semigroup on L2 (ℝ) with suitable upper bounds on its heat kernels, and L has a bounded holomorphic functional calculus on L2 (ℝ). In this article, we introduce new function spaces HL1 (ℝ × ℝ) and BMOL(ℝ × ℝ) (dual to the space HL*1(ℝ × ℝ) in which L* is the adjoint operator of L) associated with L, and they generalize the classical Hardy and BMO spaces on product domains. We obtain a molecular decomposition of function for HL1(ℝ × ℝ) by using the theory of tent spaces and establish a characterization of BMOL (ℝ × ℝ) in terms of Carleson conditions. We also show that the John-Nirenberg inequality holds for the space BMOL (ℝ × ℝ). Applications include large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form or nondivergence form in one dimension.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c3b2882e28672d36841017d783942cc7c1974570

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In Applied Mathematics

1. Dachun Yang
2. Xuan Thinh Duong
3. Sibei Yang
4. Ji Li
5. Lixin Yan
6. S. Hofmann
7. Renjin Jiang
8. P. Auscher

Citations in the corpus

• Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications (2012, background): “Based on this, Duong and Yan [36, 37] introduced the BMO-type space BMOL(R n) associated with L and proved that the dual space of H1 L(R n) is just BMOL∗(R n), where L∗ denotes the adjoint operator of L in L2(Rn).”
• Localized BMO and BLO Spaces on RD-Spaces and Applications to Schr\ (2009, background): “[16] Y. Han, D. Müller and D. Yang, A Theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr.”
• Orlicz-Hardy spaces associated with operators (2009, methodology): “n) := �S L(f )� L1(Rn) 6 ] studied H 1(R n ) and its duality.”
• Localized Hardy Spaces $H^1$ Related to Admissible Functions on RD-Spaces and Applications to Schr\ (2009, background): “In particular, Fefferman [17], Shen [47] and Zhong [58] established some basic results, including estimates of the fundamental solutions and the boundedness on Lebesgue spaces of Riesz transforms, for the Schrödinger 2000 Mathematics Subject Classification.”
• BMO spaces associated with semigroups of operators (2011, background): “n the classical analysis. One of the related subjects is Fefferman’s H1 − BMO duality theory which is missing in the context of semigroups in general. In a recent remarkable work of Doung and Yan (see [DY]), a H1 − BMO duality theory was proved for general Markov semigroups instead of the classical P”
• COMPARISON OF THE CLASSICAL BMO WITH THE BMO SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS (2006, background): “We refer the reader to Section 6 of [16] for a discussion on the dimension of the space KL of BMOL(R) when L is a second order divergence form elliptic operator or a Schrödinger operator.”
• New Abstract Hardy Spaces (2007, methodology): “2 of [17]) : a function m is called a H(1) L-molecule associated to a ball Q ⊂ R if”
• Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces (2010, methodology): “For more general operators L whose heat kernel satisfies a pointwise Gaussian upper bound, an adapted Hardy space theory has been introduced by Auscher, Duong and McIntosh [9], and by Duong and Yan, [26], [27].”
• Hardy and BMO spaces associated to divergence form elliptic operators (2006, neutral): “We should mention that some of the problems discussed above were earlier considered under different restrictions on the operator and underlying space, but always assuming pointwise bounds for the heat semigroup (see [8], [17], [18]).”
• Orlicz-Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates (2009, background): “In recent years, the study of Hardy spaces and BMO spaces associated with different operators inspired great interests; see, for example, [1, 2, 3, 11, 12, 13, 14, 18, 17, 21, 33] and their references.”
• Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates (2011, background): “We then specialize, in Section 7, to the case that the heat kernel enjoys a pointwise Gaussian bound, to prove some sharper results in the spirit of [ DY2 ].”
• Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates (2013, background): “The theory on Hardy spaces associated with such operators L was investigated in [5, 30].”
• Calderón reproducing formulas and new Besov spaces associated with operators (2012, methodology): “We will develop the theme in the works [17,18,1,2] and define the Besov norm Ḃ p,q via quadratic norms of L.”
• Endpoint estimates for Riesz transforms of magnetic Schrödinger operators (2006, background): “3 in [11] that ‖πA(a)‖L2≤c|||a|||T 2 2 ≤c|B|−1/2.”
• Smooth Fourier multipliers on group von Neumann algebras (2010, background): “Duong and Yan [10, 11] —see also [1, 20]— extended it to certain semigroups of positive maps, but still imposing the existence of a metric in the underlying space, something that a priori we do not have at our disposal.”