Banach Journal of Mathematical Analysis

Poisson semigroup, area function, and the characterization of Hardy space associated to degenerate Schrödinger operators

Abstract

Let

$\begin{eqnarray*}Lf(x)=-\frac{1}{\omega(x)}\sum_{i,j}\partial_{i}(a_{ij}(\cdot)\partial _{j}f)(x)+V(x)f(x)\end{eqnarray*}$ be the degenerate Schrödinger operator, where $\omega$ is a weight from the Muckenhoupt class $A_{2}$ and $V$ is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure $\omega(x)dx$. Based on some smoothness estimates of the Poisson semigroup $e^{-t\sqrt{L}}$, we introduce the area function $S^{L}_{P}$ associated with $e^{-t\sqrt{L}}$ to characterize the Hardy space associated with $L$.

Article information

Source
Banach J. Math. Anal., Volume 10, Number 4 (2016), 727-749.

Dates
Accepted: 6 January 2016
First available in Project Euclid: 31 August 2016

https://projecteuclid.org/euclid.bjma/1472657854

Digital Object Identifier
doi:10.1215/17358787-3649986

Mathematical Reviews number (MathSciNet)
MR3543909

Zentralblatt MATH identifier
1347.42037

Citation

Huang, Jizheng; Li, Pengtao; Liu, Yu. Poisson semigroup, area function, and the characterization of Hardy space associated to degenerate Schrödinger operators. Banach J. Math. Anal. 10 (2016), no. 4, 727--749. doi:10.1215/17358787-3649986. https://projecteuclid.org/euclid.bjma/1472657854

References

• [1] J. Cao and D. Yang, Hardy spaces $H^{p}_{L}(\mathbb{R}^{n})$ associated with operators satisfying $k$-Davies–Gaffney estimates, Sci. China Math. 55 (2012), no. 7, 1403–1440.
• [2] R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335.
• [3] X. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943–973.
• [4] J. Dziubański, Note on $H^{1}$ spaces related to degenerate Schrödinger operators, Illinois J. Math. 49 (2005), no. 4, 1271–1297.
• [5] J. Dziubański and J. Zienkiewicz, Hardy space $H^{1}$ associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoam. 15 (1999), no. 2, 279–296.
• [6] J. Dziubański and J. Zienkiewicz, $H^{p}$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes, Colloq. Math. 98 (2003), no. 1, 5–38.
• [7] C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3–4, 137–193.
• [8] G. Folland and E. M. Stein, Hardy Spaces on Homogeneous Group, Math. Notes 28, Princeton Univ. Press, Princeton, 1982.
• [9] W. Hebisch and L. Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 5, 1437–1481.
• [10] 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.
• [11] R. Jiang and D. Yang, Orlicz–Hardy spaces associated with operators, Sci. China Ser. A 52 (2009), no. 5, 1042–1080.
• [12] R. Jiang and D. Yang, Orlicz–Hardy spaces associated with operators satisfying Davies–Gaffney estimates, Commun. Contemp. Math. 13 (2011), no. 2, 331–373.
• [13] K. Kurata and S. Sugano, Fundamental solution, eigenvalue asymptotics and eigenfunctions of degenerate elliptic operators with positive potentials, Studia Math. 138 (2000), no. 2, 101–119.
• [14] C. Lin, H. Liu, and Y. Liu, Hardy spaces associated with Schrödinger operators on the Heisenberg group, preprint, arXiv:1106.4960v1 [math.AP].
• [15] L. E. Persson, M. Ragusa, N. Samko, and P. Wall, “Commutators of Hardy operators in vanishing Morrey spaces” in Conference Proceedings (ICNPAA, 2012), Amer. Inst. Phys. (AIP) 1493 (2012), no. 1, 859–866.
• [16] S. Polidoro and M. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam. 24 (2008), no. 3, 1011–1046.
• [17] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, NJ, 1993.
• [18] D. Yang, D. Yang, and Y. Zhou, Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrödinger operators, Potential Anal. 30 (2009), no. 3, 271–300.
• [19] D. Yang, D. Yang, and Y. Zhou, Localized $\mathit{BMO}$ and $\mathit{BLO}$ spaces on $\mathit{RD}$-spaces and applications to Schrödinger operators, Commun. Pure Appl. Anal. 9 (2010), no. 3, 779–812.
• [20] D. Yang, D. Yang, and Y. Zhou, Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators, Nagoya Math. J. 198 (2010), 77–119.
• [21] D. Yang and Y. Zhou, Localized Hardy spaces $H^{1}$ related to admissible functions on $\mathit{RD}$-spaces and applications to Schrödinger operators, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1197–1239.