| Title:
|
Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators (English) |
| Author:
|
Yang, Sibei |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
3 |
| Year:
|
2015 |
| Pages:
|
747-779 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $L:=-\Delta +V$ be a Schrödinger operator on $\mathbb {R}^n$ with $n\ge 3$ and $V\ge 0$ satisfying $\Delta ^{-1} V\in L^\infty (\mathbb {R}^n)$. Assume that $\varphi \colon \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,\cdot )$ is an Orlicz function, $\varphi (\cdot ,t)\in {\mathbb A}_{\infty }(\mathbb {R}^n)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on $\mathbb {R}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}(\mathbb {R}^n)\ni f\mapsto wf\in H_\varphi (\mathbb {R}^n)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}(\mathbb {R}^n)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }(\mathbb {R}^n)$ under some assumptions on $\varphi $. As applications, the author further obtains the atomic and molecular characterizations of the space $H_{\varphi ,L}(\mathbb {R}^n)$ associated with $w$, and proves that the operator $(-\Delta )^{-1/2}L^{1/2}$ is an isomorphism of the spaces $H_{\varphi ,L}(\mathbb {R}^n)$ and $H_{\varphi }(\mathbb {R}^n)$. All these results are new even when $\varphi (x,t):=t^p$, for all $x\in \mathbb {R}^n$ and $t\in [0,\infty )$, with $p\in ({n}/{(n+\mu _0)},1)$ and some $\mu _0\in (0,1]$. (English) |
| Keyword:
|
Musielak-Orlicz-Hardy space |
| Keyword:
|
Schrödinger operator |
| Keyword:
|
$L$-harmonic function |
| Keyword:
|
isomorphism of Hardy space |
| Keyword:
|
atom |
| Keyword:
|
molecule |
| MSC:
|
35J10 |
| MSC:
|
42B20 |
| MSC:
|
42B30 |
| MSC:
|
42B35 |
| MSC:
|
42B37 |
| MSC:
|
46E30 |
| idZBL:
|
Zbl 06537690 |
| idMR:
|
MR3407603 |
| DOI:
|
10.1007/s10587-015-0206-1 |
| . |
| Date available:
|
2015-10-04T18:16:22Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144441 |
| . |
| Reference:
|
[1] Bonami, A., Grellier, S., Ky, L. D.: Paraproducts and products of functions in BMO$(\mathbb R^n)$ and ${\cal H}^1(\mathbb R^n)$ through wavelets.J. Math. Pures Appl. (9) 97 (2012), 230-241 French summary. MR 2887623, 10.1016/j.matpur.2011.06.002 |
| Reference:
|
[2] Bonami, A., Iwaniec, T., Jones, P., Zinsmeister, M.: On the product of functions in BMO and $H^1$.Ann. Inst. Fourier 57 (2007), 1405-1439. Zbl 1132.42010, MR 2364134 |
| Reference:
|
[3] Bui, T. A., Cao, J., Ky, L. D., Yang, D., Yang, S.: Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates.Anal. Geom. Metr. Spaces (electronic only) 1 (2013), 69-129. Zbl 1261.42034, MR 3108869, 10.2478/agms-2012-0006 |
| Reference:
|
[4] Cao, J., Chang, D.-C., Yang, D., Yang, S.: Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces.Commun. Pure Appl. Anal. 13 (2014), 1435-1463. MR 3177739, 10.3934/cpaa.2014.13.1435 |
| Reference:
|
[5] Duong, X. T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds.J. Am. Math. Soc. 18 (2005), 943-973. Zbl 1078.42013, MR 2163867, 10.1090/S0894-0347-05-00496-0 |
| Reference:
|
[6] Dziubański, J., Zienkiewicz, J.: A characterization of Hardy spaces associated with certain Schrödinger operators.Potential Anal. 41 (2014), 917-930. Zbl 1301.42039, MR 3264827, 10.1007/s11118-014-9400-2 |
| Reference:
|
[7] Dziubański, J., Zienkiewicz, J.: On isomorphisms of Hardy spaces associated with Schrödinger operators.J. Fourier Anal. Appl. 19 (2013), 447-456. Zbl 1305.42025, MR 3048584, 10.1007/s00041-013-9262-9 |
| Reference:
|
[8] Fefferman, C. L., Stein, E. M.: $H^p$ spaces of several variables.Acta Math. 129 (1972), 137-193. MR 0447953, 10.1007/BF02392215 |
| Reference:
|
[9] García-Cuerva, J., Francia, J. L. Rubio de: Weighted Norm Inequalities and Related Topics.North-Holland Mathematics Studies 116 North-Holland, Amsterdam (1985). MR 0807149 |
| Reference:
|
[10] Grafakos, L.: Modern Fourier Analysis.Graduate Texts in Mathematics 250 Springer, New York (2009). Zbl 1158.42001, MR 2463316 |
| Reference:
|
[11] 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. 1007 (2011), 78 pages. Zbl 1232.42018, MR 2868142 |
| Reference:
|
[12] 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:
|
[13] Hofmann, S., Mayboroda, S., McIntosh, A.: Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces.Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 723-800 French summary. Zbl 1243.47072, MR 2931518, 10.24033/asens.2154 |
| Reference:
|
[14] Hou, S., Yang, D., Yang, S.: Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications.Commun. Contemp. Math. 15 (2013), Article ID1350029, 37 pages. Zbl 1285.42020, MR 3139410 |
| Reference:
|
[15] Janson, S.: Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation.Duke Math. J. 47 (1980), 959-982. Zbl 0453.46027, MR 0596123, 10.1215/S0012-7094-80-04755-9 |
| Reference:
|
[16] Jiang, R., Yang, D.: Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates.Commun. Contemp. Math. 13 (2011), 331-373. Zbl 1221.42042, MR 2794490, 10.1142/S0219199711004221 |
| Reference:
|
[17] Jiang, R., Yang, D.: New Orlicz-Hardy spaces associated with divergence form elliptic operators.J. Funct. Anal. 258 (2010), 1167-1224. Zbl 1205.46014, MR 2565837, 10.1016/j.jfa.2009.10.018 |
| Reference:
|
[18] Jiang, R., Yang, D., Yang, D.: Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators.Forum Math. 24 (2012), 471-494. Zbl 1248.42023, MR 2926631, 10.1515/form.2011.067 |
| Reference:
|
[19] Ky, L. D.: Endpoint estimates for commutators of singular integrals related to Schrödinger operators.To appear in Rev. Mat. Iberoam. |
| Reference:
|
[20] Ky, L. D.: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators.Integral Equations Oper. Theory 78 (2014), 115-150. Zbl 1284.42073, MR 3147406, 10.1007/s00020-013-2111-z |
| Reference:
|
[21] Ky, L. D.: Bilinear decompositions and commutators of singular integral operators.Trans. Am. Math. Soc. 365 (2013), 2931-2958. Zbl 1272.42010, MR 3034454 |
| Reference:
|
[22] Musielak, J.: Orlicz Spaces and Modular Spaces.Lecture Notes in Mathematics 1034 Springer, Berlin (1983). Zbl 0557.46020, MR 0724434 |
| Reference:
|
[23] 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:
|
[24] Rao, M. M., Ren, Z. D.: Theory of Orlicz Spaces.Pure and Applied Mathematics 146 Marcel Dekker, New York (1991). Zbl 0724.46032, MR 1113700 |
| Reference:
|
[25] Semenov, Y. A.: Stability of $L^p$-spectrum of generalized Schrödinger operators and equivalence of Green's functions.Int. Math. Res. Not. 12 (1997), 573-593. Zbl 0905.47031, MR 1456565, 10.1155/S107379289700038X |
| Reference:
|
[26] Simon, B.: Functional Integration and Quantum Physics.AMS Chelsea Publishing, Providence (2005). Zbl 1061.28010, MR 2105995 |
| Reference:
|
[27] Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces.Indiana Univ. Math. J. 28 (1979), 511-544. MR 0529683, 10.1512/iumj.1979.28.28037 |
| Reference:
|
[28] Strömberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces.Lecture Notes in Mathematics 1381 Springer, Berlin (1989). Zbl 0676.42021, MR 1011673, 10.1007/BFb0091160 |
| Reference:
|
[29] Yan, L.: Classes of Hardy spaces associated with operators, duality theorem and applications.Trans. Am. Math. Soc. 360 (2008), 4383-4408. Zbl 1273.42022, MR 2395177, 10.1090/S0002-9947-08-04476-0 |
| Reference:
|
[30] Yang, D., Yang, S.: Musielak-Orlicz Hardy spaces associated with operators and their applications.J. Geom. Anal. 24 (2014), 495-570. Zbl 1302.42033, MR 3145932, 10.1007/s12220-012-9344-y |
| Reference:
|
[31] Yang, D., Yang, S.: Local Hardy spaces of Musielak-Orlicz type and their applications.Sci. China Math. 55 (2012), 1677-1720. Zbl 1266.42055, MR 2955251, 10.1007/s11425-012-4377-z |
| . |
