Taiwanese Journal of Mathematics

SOME ESTIMATES FOR SCHRÖDINGER TYPE OPERATORS ON MUSIELAK-ORLICZ-HARDY SPACES

Sibei Yang

Full-text: Open access

Abstract

Let $L:=-\mathrm{div}(A\nabla)+V$ be a Schrödinger type operator with the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q_0}(\mathbb{R}^n)$ for some $q_0\in[n,\infty)$ with $n\ge3$, where $A$ satisfies the uniformly elliptic condition. Assume that $\varphi:\,\mathbb{R}^n\times[0,\infty)\to[0,\infty)$ 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) and its uniformly critical lower type index $i(\varphi)\in(\frac{n}{n+\alpha_0},1]$, where $\alpha_0\in(0,1]$ measures the regularity of kernels of the semigroup generalized by $L_0:=-\mathrm{div}(A\nabla)$. In this article, we first prove that operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi,\,L}(\mathbb{R}^n)$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, we also obtain the boundedness of $VL^{-1}$ and $\nabla^2L^{-1}$ on $H_{\varphi,\,L}(\mathbb{R}^n)$. All these results are new even when $\varphi(x,t):=t^p$, with $p\in(\frac{n}{n+\alpha_0},1]$, for all $x\in\mathbb{R}^n$ and $t\in[0,\infty)$.

Article information

Source
Taiwanese J. Math., Volume 18, Number 4 (2014), 1293-1328.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706491

Digital Object Identifier
doi:10.11650/tjm.18.2014.3897

Mathematical Reviews number (MathSciNet)
MR3245444

Zentralblatt MATH identifier
1357.42013

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis 42B25: Maximal functions, Littlewood-Paley theory 35J10: Schrödinger operator [See also 35Pxx] 42B37: Harmonic analysis and PDE [See also 35-XX] 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords
Musielak-Orlicz-Hardy space Schrödinger type operator Lusin area function atom second order Riesz transform fundamental solution

Citation

Yang, Sibei. SOME ESTIMATES FOR SCHRÖDINGER TYPE OPERATORS ON MUSIELAK-ORLICZ-HARDY SPACES. Taiwanese J. Math. 18 (2014), no. 4, 1293--1328. doi:10.11650/tjm.18.2014.3897. https://projecteuclid.org/euclid.twjm/1499706491


Export citation

References

  • P. Auscher and B. Ben Ali, Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier $($Grenoble$)$, 57 (2007), 1975-2013.
  • P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach Space Valued Singular Integral Operators and Hardy Spaces, Unpublished Manuscript, 2005.
  • P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., 18 (2008), 192-248.
  • P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque, 249 (1998), viii+172,pp.
  • M. Avellaneda and F. Lin, $L^p$ bounds on singular in homogenization, Comm. Pure Appl. Math., 44 (1991), 897-910.
  • N. Badr and B. Ben Ali, $L^p$ boundedness of the Riesz transform related to Schrödinger operators on a manifold, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8(5) (2009), 725-765.
  • A. Bonami, J. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat., 54 (2010), 341-358.
  • A. Bonami, S. Grellier and L. D. Ky, Paraproducts and products of functions in $BMO(\rn)$ and $H^1(\rn)$ through wavelets, J. Math. Pures Appl. (9), 97 (2012), 230-241.
  • A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in $BMO$ and $H^1$, Ann. Inst. Fourier $($Grenoble$)$, 57 (2007), 1405-1439.
  • J. Cao, D.-C. Chang, D. Yang and S. Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Comm. Pure Appl. Anal., 13 (2014), 1435-1463.
  • R. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), 72 (1993), 247-286.
  • X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973.
  • J. Dziubański, Note on $H^1$ spaces related to degenerate Schrödinger operators, Illinois J. Math., 49 (2005), 1271-1297.
  • J. Dziubański and J. Zienkiewicz, $H^p$ spaces for Schrödinger operators, Fourier analysis and related topics, Banach Center Publ., 56, Polish Acad. Sci., Warsaw, 2002, 45-53.
  • J. Dziubański and J. Zienkiewicz, $H^p$ spaces associated with Schrödinger operators with potential from reverse Hölder classes, Colloq. Math., 98 (2003), 5-38.
  • C. Fefferman, The uncertainly principle, Bull. Amer. Math. Soc., 9 (1983), 129-206.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.
  • J. Garc\ptmrsía-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. $($Rozprawy Mat.$)$, 162 (1979), 1-63.
  • J. Garc\ptmrsía-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985.
  • F. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277.
  • L. Grafakos, Modern Fourier Analysis, 2nd edition, Graduate Texts in Mathematics 250, Springer, New York, 2009.
  • S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, Commun. Contemp. Math., 15(6) (2013), 1350029, 37 pp.
  • S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc., 214 (2011), no. 1007, vi+78 pp.
  • S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann., 344 (2009), 37-116.
  • S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces, Ann. Sci. École Norm. Sup. (4), 44 (2011), 723-800.
  • S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J., 47 (1980), 959-982.
  • R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal., 258 (2010), 1167-1224.
  • R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math., 13 (2011), 331-373.
  • R. Jiang, Da. Yang and Do. Yang, Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators, Forum Math., 24 (2012), 471-494.
  • R. Jiang, D. Yang and Y. Zhou, Orlicz-Hardy spaces associated with operators, Sci. China Ser. A, 52 (2009), 1042-1080.
  • K. Kurata and S. Sugano, A remark on estimates for uniformly elliptic operators on weighted $L^p$ spaces and Morrey spaces, Math. Nachr., 209 (2000), 137-150.
  • L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory, 78 (2014), 115-150.
  • J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983.
  • E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan, 37 (1985), 207-218.
  • E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, N. J., 2005.
  • M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.
  • Z. Shen, $L^p$ estimates for Schrödinger operators with certain potential, Ann. Inst. Fourier $($Grenoble$)$, 45 (1995), 513-546.
  • H. Smith, Parametrix construction for a class of subelliptic differential operators, Duke Math. J., 63 (1991), 343-354.
  • E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^p$-spaces, Acta Math., 103 (1960), 25-62.
  • J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J., 28 (1979), 511-544.
  • J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer-Verlag, Berlin, 1989.
  • L. Tang, Weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators, arXiv: 1109.0100.
  • L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc., 360 (2008), 4383-4408.
  • D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications, Sci. China Math., 55 (2012), 1677-1720.
  • D. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators and their applications, J. Geom. Anal., 24 (2014), 495-570.
  • J. Zhong, The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line, 3 (1999), 1-48.