Abstract
In this chapter, we study approximations of the identity on \({\mathbb{R}}^{D}\) with the measures μ satisfying (0.0.1). To this end, we first introduce an important notion of coefficients δ(Q, R) for cubes Q and R in \({\mathbb{R}}^{D}\). It turns out that δ(Q, R) characterizes the geometric relationship between Q and R. Using this notion, we further study cubes of different generations in terms of δ(Q, R), which are versions of dyadic cubes in the setting \(({\mathbb{R}}^{D},\vert \cdot \vert,\mu )\). Then we construct the functions, \(\{f_{y,\,k}\}_{y\in \mathrm{\,supp\,}\mu,\ k\in \mathbb{Z}}\), which originate the kernels of approximations of the identity. Via these functions, we introduce and establish some important properties of the approximations of the identity on \({\mathbb{R}}^{D}\).
Keywords
- Dyadic Cubes
- Geometric Relationship
- Double Cube
- Initial Cube
- Sense Modulo
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
See, for example, [40].
References
D. Deng, Y. Han, D. Yang, Besov spaces with non-doubling measures. Trans. Am. Math. Soc. 358, 2965–3001 (2006)
L. Grafakos, Classical Fourier Analysis (Springer, New York, 2008)
F. Nazarov, S. Treil, A. Volberg, The Tb-theorem on non-homogeneous spaces. Acta Math. 190, 151–239 (2003)
X. Tolsa, BMO, H 1 and Calderón–Zygmund operators for non doubling measures. Math. Ann. 319, 89–149 (2001)
X. Tolsa, Littlewood–Paley theory and the T(1) theorem with non-doubling measures. Adv. Math. 164, 57–116 (2001)
X. Tolsa, The space H 1 for nondoubling measures in terms of a grand maximal operator. Trans. Am. Math. Soc. 355, 315–348 (2003)
X. Tolsa, Characterization of the atomic space H 1 for non doubling measures in terms of a grand maximal operator. http://mat.uab.cat/~xtolsa/hh.pdf
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Yang, D., Yang, D., Hu, G. (2013). Approximations of the Identity. In: The Hardy Space H1 with Non-doubling Measures and Their Applications. Lecture Notes in Mathematics, vol 2084. Springer, Cham. https://doi.org/10.1007/978-3-319-00825-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-00825-7_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00824-0
Online ISBN: 978-3-319-00825-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
