Geodesic
Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 747-779
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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
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 $0$, 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 $\mu _0\in (0,1]$.
DOI : 10.1007/s10587-015-0206-1
Classification : 35J10, 42B20, 42B30, 42B35, 42B37, 46E30
Keywords: Musielak-Orlicz-Hardy space; Schrödinger operator; $L$-harmonic function; isomorphism of Hardy space; atom; molecule 
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     title = {Isomorphisms and several characterizations of {Musielak-Orlicz-Hardy} spaces associated with some {Schr\"odinger} operators},
     journal = {Czechoslovak Mathematical Journal},
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     year = {2015},
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Yang, Sibei. Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 747-779. doi: 10.1007/s10587-015-0206-1

[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. | DOI | MR

[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. | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR

[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. | DOI | MR | Zbl

[6] Dziubański, J., Zienkiewicz, J.: A characterization of Hardy spaces associated with certain Schrödinger operators. Potential Anal. 41 (2014), 917-930. | DOI | MR | Zbl

[7] Dziubański, J., Zienkiewicz, J.: On isomorphisms of Hardy spaces associated with Schrödinger operators. J. Fourier Anal. Appl. 19 (2013), 447-456. | DOI | MR | Zbl

[8] Fefferman, C. L., Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. | DOI | MR

[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

[10] Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics 250 Springer, New York (2009). | MR | Zbl

[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. | MR | Zbl

[12] Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344 (2009), 37-116. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | MR | Zbl

[15] Janson, S.: Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J. 47 (1980), 959-982. | DOI | MR | Zbl

[16] Jiang, R., Yang, D.: Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun. Contemp. Math. 13 (2011), 331-373. | DOI | MR | Zbl

[17] Jiang, R., Yang, D.: New Orlicz-Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal. 258 (2010), 1167-1224. | DOI | MR | Zbl

[18] Jiang, R., Yang, D., Yang, D.: Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators. Forum Math. 24 (2012), 471-494. | DOI | MR | Zbl

[19] Ky, L. D.: Endpoint estimates for commutators of singular integrals related to Schrödinger operators. To appear in Rev. Mat. Iberoam.

[20] Ky, L. D.: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Oper. Theory 78 (2014), 115-150. | DOI | MR | Zbl

[21] Ky, L. D.: Bilinear decompositions and commutators of singular integral operators. Trans. Am. Math. Soc. 365 (2013), 2931-2958. | MR | Zbl

[22] Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034 Springer, Berlin (1983). | MR | Zbl

[23] Ouhabaz, E. M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series 31 Princeton University Press, Princeton (2005). | MR | Zbl

[24] Rao, M. M., Ren, Z. D.: Theory of Orlicz Spaces. Pure and Applied Mathematics 146 Marcel Dekker, New York (1991). | MR | Zbl

[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. | DOI | MR | Zbl

[26] Simon, B.: Functional Integration and Quantum Physics. AMS Chelsea Publishing, Providence (2005). | MR | Zbl

[27] Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28 (1979), 511-544. | DOI | MR

[28] Strömberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Mathematics 1381 Springer, Berlin (1989). | DOI | MR | Zbl

[29] Yan, L.: Classes of Hardy spaces associated with operators, duality theorem and applications. Trans. Am. Math. Soc. 360 (2008), 4383-4408. | DOI | MR | Zbl

[30] Yang, D., Yang, S.: Musielak-Orlicz Hardy spaces associated with operators and their applications. J. Geom. Anal. 24 (2014), 495-570. | DOI | MR | Zbl

[31] Yang, D., Yang, S.: Local Hardy spaces of Musielak-Orlicz type and their applications. Sci. China Math. 55 (2012), 1677-1720. | DOI | MR | Zbl

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