Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schr ödinger Operators

Yang, Dachun ; Yang, Sibei

Canadian mathematical bulletin, Tome 58 (2015) no. 2, pp. 432-448

Résumé

Let $A\,:=\,-\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,.\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,+\,V$ be a magnetic Schrödinger operator on ${{\mathbb{R}}^{n}}$ , where $$\vec{a}:=\left( {{a}_{1}},...,{{a}_{n}} \right)\in L_{\text{loc}}^{2}\left( {{\mathbb{R}}^{n}},{{\mathbb{R}}^{n}} \right)\operatorname{and}0\le V\in L_{\text{loc}}^{1}\left( {{\mathbb{R}}^{n}} \right)$$ satisfy some reverse Hölder conditions. Let $\phi :\,{{\mathbb{R}}^{n}}\,\times \,[0,\,\infty )\,\to \,[0,\,\infty )$ be such that $\phi \left( x,\,. \right)$ for any given $x\,\in \,{{\mathbb{R}}^{n}}$ is an Orlicz function, $\phi \left( ^{.}\,,\,t \right)\,\in \,{{\mathbb{A}}_{\infty }}\left( {{\mathbb{R}}^{n}} \right)$ for all $t\,\in \,\left( 0,\,\infty\right)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I\left( \phi\right)\,\in \,(0,\,1]$ . In this article, the authors prove that second-order Riesz transforms $V{{A}^{-1}}$ and ${{\left( \nabla \,-\,i\overrightarrow{a} \right)}^{2}}{{A}^{-1}}$ are bounded from the Musielak–Orlicz–Hardy space ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$ , associated with $A$ , to theMusielak–Orlicz space ${{L}^{\phi }}\left( {{\mathbb{R}}^{n}} \right)$ . Moreover, we establish the boundedness of $V{{A}^{-1}}$ on ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$ . As applications, some maximal inequalities associated with $A$ in the scale of ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$ are obtained.

Détail

DOI : 10.4153/CMB-2014-060-x

Mots-clés : 42B30, 42B35, 42B25, 35J10, 42B37, 46E30, Musielak–Orlicz–Hardy space, magnetic Schrödinger operator, atom, second-order Riesz transform, maximal inequality

Bibliographie

[1] [1] Ben Ali, B., Maximal inequalities and Riesz transform estimates on L? spaces for magnetic Schrodinger operators I. J. Funct. Anal. 259 (2010), no. 7,1631–1672. http://dx.doi.Org/10.1016/j.jfa.2009.09.003 Google Scholar

[2] [2] Ben Ali, B., Maximal inequalities and Riesz transform estimates on L? spaces for magnetic Schrodinger operators II. Math. Z. 274 (2013), no. 1–2, 85–116. Google Scholar | DOI

[3] [3] Bui, T. A., Cao, J., Ky, L. D., Yang, D., and Yang, S., Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal. Geom. Metr. Spaces 1 (2013), 69–129. Google Scholar

[4] [4] Bui, T. A. and J. Li, Orlicz-Hardy spaces associated to operators satisfying bounded Hfunctional calculus and Davies-Gaffney estimates. J. Math. Anal. Appl. 373 (2011), no. 2, 485–501. http://dx.doi.Org/10.1016/j.jmaa.2O10.07.050 Google Scholar

[5] [5] Cao, J., Chang, D.-C., Yang, D., and Yang, S., Boundedness of second order Riesz transforms associated to Schrodinger operators on Musielak-Orlicz-Hardy spaces. Commun. Pure Appl. Anal. 13 (2014), no. 4, 1435–1463. Google Scholar | DOI

[6] [6] Cao, J., Chang, D.-C., Yang, D., and Yang, S., Estimates for second order Riesz transforms associated with magnetic Schrodinger operators on Musielak-Orlicz-Hardy spaces. Appl. Anal. 93 (2014), no. 11, 2519–2545. Google Scholar | DOI

[7] [7] Cruz-Uribe, D. and Neugebauer, C. J., The structure of the reverse Holder classes. Trans. Amer. Math. Soc. 347 (1995), 2941–2960. Google Scholar

[8] [8] Duong, X. T. and Li, J., Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. J. Funct. Anal. 264 (2013), 1409–1437. http://dx.doi.Org/10.1016/j.jfa.2013.01.006 Google Scholar

[9] [9] Duong, X. T., Xiao, J., and Yan, L., Old and new Morrey spaces with heat kernel bounds. J. Fourier Anal. Appl. 13 (2007), 87–111. Google Scholar | DOI

[10] [10] Duong, X. T. and Yan, L., Commutators of Riesz transforms of magnetic Schrodinger operators. ManuscriptaMath. 127 (2008), 219–234. Google Scholar | DOI

[11] [11] Fefferman, C. and Stein, E. M., Hp spaces of several variables. Acta Math. 129 (1972), 137–193. Google Scholar | DOI

[12] [12] Gehring, F., The Lp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265–277. Google Scholar | DOI

[13] [13] Grafakos, L., Modern Fourier analysis. Second éd., Graduate Texts in Mathematics. 250, Springer, New York, 2009. Google Scholar

[14] [14] Hou, S., Yang, D., and Yang, S., Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications. Commun. Contemp. Math. 15 (2013), no. 6, 1350029. http://dx.doi.Org/10.1142/S0219199713500296 Google Scholar

[15] [15] Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., and Yan, L., Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Amer. Math. Soc. 214 (2011), no. 1007. Google Scholar

[16] [16] Jiang, R. and Yang, D., New Orlicz-Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal. 258 (2010), 1167–1224. http://dx.doi.Org/10.1016/j.jfa.2009.10.018 Google Scholar

[17] [17] Jiang, R. and Yang, D., Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun. Contemp. Math. 13 (2011), 331–373. http://dx.doi.Org/10.1142/S0219199711004221 Google Scholar

[18] [18] Jiang, R., Yang, Da., and Yang, Do., Maximal function characterizations of Hardy spaces associated with magnetic Schrodinger operators. Forum Math. 24 (2012), 471–494. Google Scholar

[19] [19] Johnson, R. and Neugebauer, C. J., Homeomorphismspreserving Ap. Rev. Mat. Iberoam. 3 (1987), 249–273. Google Scholar | DOI

[20] [20] Kurata, K., An estimate on the heat kernel of magnetic Schrodinger operators and uniformly elliptic operators with non-negative potentials. J. London Math. Soc. (2) 62 (2000), 885–903. http://dx.doi.Org/10.1112/S002461070000137X Google Scholar

[21] [21] Kurata, K. and Sugano, S., Estimates of the fundamental solution for magnetic Schrodinger operators and their applications. Tohoku Math. J. (2) 52 (2000), 367–382. http://dx.doi.Org/10.2748/tmj/1178207819 Google Scholar

[22] [22] Ky, L. D., Bilinear decompositions and commutators of singular integral operators. Trans. Amer. Math. Soc. 365 (2013), 2931–2958. Google Scholar

[23] [23] Ky, L. D., New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Operator Theory 78 (2014), 115–150. Google Scholar

[24] [24] Musielak, J., Orlicz spaces and modular spaces. Lecture Notes in Mathematics, 1034, Springer-Verlag, Berlin, 1983. Google Scholar

[25] [25] Rao, M. M. and Ren, Z. D., Theory of Orlicz spaces. Marcel Dekker, Inc., New York, 1991. Google Scholar

[26] [26] Shen, Z., Estimates in L? for magnetic Schrodinger operators. Indiana Univ. Math. J. 45 (1996), 817–841. Google Scholar

[27] [27] Shen, Z., Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrodinger operators with magnetic fields. Trans. Amer. Math. Soc. 348 (1996), 4465–4488. Google Scholar | DOI

[28] [28] Shen, Z., LP estimates for Schrodinger operators with certain potential. Ann. Inst. Fourier (Grenoble) 45 (1995), 513–546. Google Scholar | DOI

[29] [29] Strômberg, J.-O. and Torchinsky, A., Weighted Hardy spaces. Lecture Notes in Mathematics, 1381, Springer-Verlag, Berlin, 1989. Google Scholar

[30] [30] Yang, Da. and Yang, Do., Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrodinger operators. Front. Math. China (to appear). Google Scholar

[31] [31] Yang, D. and Yang, S., Musielak-Orlicz Hardy spaces associated with operators and their applications. J. Geom. Anal. 24 (2014), 495–570. Google Scholar | DOI