| Title:
|
Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces (English) |
| Author:
|
Yan, Xuefang |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
61-82 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(X, d, \mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu $. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2(X)$. Assume that the semigroup ${\rm e}^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H^p_L(X)$ be the Hardy space associated with $L.$ We show the boundedness of Stein's square function ${\mathcal G}_{\delta }(L)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H^p_L(X)$ to $L^{p}(X)$, and also study the boundedness of Bochner-Riesz means on Hardy spaces $H^p_L(X)$ for $0<p\leq 1$. (English) |
| Keyword:
|
non-negative self-adjoint operator |
| Keyword:
|
Stein's square function |
| Keyword:
|
Bochner-Riesz means |
| Keyword:
|
Davies-Gaffney estimate |
| Keyword:
|
molecule Hardy space |
| MSC:
|
42B15 |
| MSC:
|
42B25 |
| MSC:
|
47F05 |
| idZBL:
|
Zbl 06433721 |
| idMR:
|
MR3336025 |
| DOI:
|
10.1007/s10587-015-0160-y |
| . |
| Date available:
|
2015-04-01T12:20:41Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144213 |
| . |
| Reference:
|
[1] Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds.J. Geom. Anal. 18 (2008), 192-248. Zbl 1217.42043, MR 2365673, 10.1007/s12220-007-9003-x |
| Reference:
|
[2] Blunck, S., Kunstmann, P. C.: Generalized Gaussian estimates and the Legendre transform.J. Oper. Theory 53 (2005), 351-365. Zbl 1117.47020, MR 2153153 |
| Reference:
|
[3] Bui, T. A., Duong, X. T.: Boundedness of singular integrals and their commutators with BMO functions on Hardy spaces.Adv. Differ. Equ. 18 (2013), 459-494. Zbl 1275.42019, MR 3086462 |
| Reference:
|
[4] Chen, P.: Sharp spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates.Colloq. Math. 133 (2013), 51-65. Zbl 1291.42012, MR 3139415, 10.4064/cm133-1-4 |
| Reference:
|
[5] Chen, P., Duong, X. T., Yan, L.: $L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers.J. Math. Soc. Japan. 65 (2013), 389-409. Zbl 1277.42011, MR 3055591, 10.2969/jmsj/06520389 |
| Reference:
|
[6] Christ, M.: $L^p$ bounds for spectral multipliers on nilpotent groups.Trans. Am. Math. Soc. 328 (1991), 73-81. MR 1104196 |
| Reference:
|
[7] Coifman, R. R., Weiss, G.: Non-Commutative Harmonic Analysis on Certain Homogeneous Spaces. Study of Certain Singular Integrals.Lecture Notes in Mathematics 242 Springer, Berlin (1971), French. Zbl 0224.43006, MR 0499948, 10.1007/BFb0058946 |
| Reference:
|
[8] Davies, E. B.: Limits on $L^{p}$ regularity of self-adjoint elliptic operators.J. Differ. Equations 135 (1997), 83-102. MR 1434916, 10.1006/jdeq.1996.3219 |
| Reference:
|
[9] Duong, X. T., Li, J.: Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus.J. Funct. Anal. 264 (2013), 1409-1437. Zbl 1271.42033, MR 3017269, 10.1016/j.jfa.2013.01.006 |
| Reference:
|
[10] Duong, X. T., Ouhabaz, E. M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers.J. Funct. Anal. 196 (2002), 443-485. MR 1943098, 10.1016/S0022-1236(02)00009-5 |
| Reference:
|
[11] Duong, X. T., Yan, L.: Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates.J. Math. Soc. Japan. 63 (2011), 295-319. Zbl 1221.42024, MR 2752441, 10.2969/jmsj/06310295 |
| Reference:
|
[12] Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., Yan, L.: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates.Mem. Am. Math. Soc. 214 (2011), no. 1007, 78 pages. Zbl 1232.42018, MR 2868142 |
| Reference:
|
[13] Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators.Math. Ann. 344 (2009), 37-116. Zbl 1162.42012, MR 2481054, 10.1007/s00208-008-0295-3 |
| Reference:
|
[14] Igari, S.: A note on the Littlewood-Paley function $g^{\ast}(f)$.Tohoku Math. J., II. Ser. 18 (1966), 232-235. MR 0199643, 10.2748/tmj/1178243450 |
| Reference:
|
[15] Igari, S., Kuratsubo, S.: A sufficient condition for $L^p$-multipliers.Pac. J. Math. 38 (1971), 85-88. MR 0306793, 10.2140/pjm.1971.38.85 |
| Reference:
|
[16] Kaneko, M., Sunouchi, G. I.: On the Littlewood-Paley and Marcinkiewicz functions in higher dimensions.Tohoku. Math. J., II. Ser. 37 (1985), 343-365. Zbl 0579.42011, MR 0799527, 10.2748/tmj/1178228647 |
| Reference:
|
[17] Kunstmann, P. C., Uhl, M.: Spectral multiplier theorems of Hörmander type on Hardy and Lebesgue spaces.Available at http://arXiv:1209.0358v1 (2012). MR 3322756 |
| Reference:
|
[18] Ouhabaz, E. M.: Analysis of Heat Equations on Domains.London Mathematical Society Monographs Series 31 Princeton University Press, Princeton (2005). Zbl 1082.35003, MR 2124040 |
| Reference:
|
[19] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis.Academic Press New York (1980). Zbl 0459.46001, MR 0751959 |
| Reference:
|
[20] Schreieck, G., Voigt, J.: Stability of the $L_{p}$-spectrum of Schrödinger operators with form-small negative part of the potential.Functional Analysis K. D. Bierstedt et al. Proceedings of the Essen Conference, 1991. Lect. Notes Pure Appl. Math. 150 (1994), 95-105 Dekker, New York. MR 1241673 |
| Reference:
|
[21] Stein, E. M.: Localization and summability of multiple Fourier series.Acta Math. 100 (1958), 93-147. Zbl 0085.28401, MR 0105592, 10.1007/BF02559603 |
| Reference:
|
[22] Yosida, K.: Functional Analysis.Grundlehren der Mathematischen Wissenschaften 123 Springer, Berlin (1978). Zbl 0365.46001, MR 0500055 |
| . |
