Taiwanese Journal of Mathematics

Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces

Qiang Huang and Chao Zhang

Full-text: Open access


Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L} = -\Delta + V$ acting on $L^2(\mathbb{R}^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q \geq n$. Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0 \leq \lambda \lt n$ denote the Morrey space on $\mathbb{R}^{n}$. In this paper, we will show that a function $f \in L^{2,\lambda}(\mathbb{R}^{n})$ is the trace of the solution of $\mathbb{L}u := u_{t} + \mathcal{L}u = 0$, $u(x,0) = f(x)$, where $u$ satisfies a Carleson-type condition \[ \sup_{x_B,r_B} r_B^{-\lambda} \int_0^{r_B^2} \!\! \int_{B(x_B,r_B)} |\nabla u(x,t)|^2 \, dx dt \leq C \lt \infty. \] Conversely, this Carleson-type condition characterizes all the $\mathbb{L}$-carolic functions whose traces belong to the Morrey space $L^{2,\lambda}(\mathbb{R}^{n})$ for all $0 \leq \lambda \lt n$. This result extends the analogous characterization found by Fabes and Neri in [8] for the classical BMO space of John and Nirenberg.

Article information

Taiwanese J. Math., Volume 23, Number 5 (2019), 1133-1151.

Received: 21 June 2018
Revised: 9 November 2018
Accepted: 12 November 2018
First available in Project Euclid: 21 November 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B35: Function spaces arising in harmonic analysis 42B37: Harmonic analysis and PDE [See also 35-XX] 35J10: Schrödinger operator [See also 35Pxx] 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Dirichlet problem heat equation Schrödinger operators Morrey space Carleson measure reverse Hölder inequality


Huang, Qiang; Zhang, Chao. Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces. Taiwanese J. Math. 23 (2019), no. 5, 1133--1151. doi:10.11650/tjm/181106. https://projecteuclid.org/euclid.twjm/1542790913

Export citation


  • M. Dindos, C. Kenig and J. Pipher, BMO solvability and the $A_{\infty}$ condition for elliptic operators, J. Geom. Anal. 21 (2011), no. 1, 78–95.
  • X. T. Duong, J. Xiao and L. Yan, Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 13 (2007), no. 1, 87–111.
  • X. T. Duong and L. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), no. 10, 1375–1420.
  • ––––, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943–973.
  • X. T. Duong, L. Yan and C. Zhang, On characterization of Poisson integrals of Schrödinger operators with BMO traces, J. Funct. Anal. 266 (2014), no. 4, 2053–2085.
  • J. Dziubański, G. Garrigós, T. Martínez, J. L. Torrea and J. Zienkiewicz, BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality, Math. Z. 249 (2005), no. 2, 329–356.
  • E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and $L_{p,\lambda}$, Indiana Univ. Math. J. 25 (1976), no. 2, 159–170.
  • E. B. Fabes and U. Neri, Characterization of temperatures with initial data in BMO, Duke Math. J. 42 (1975), no. 4, 725–734.
  • ––––, Dirichlet problem in Lipschitz domains with BMO data, Proc. Amer. Math. Soc. 78 (1980), no. 1, 33–39.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193.
  • W. Gao and Y. Jiang, $L^p$ estimate for parabolic Schrödinger operator with certain potentials, J. Math. Anal. Appl. 310 (2005), no. 1, 128–143.
  • F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277.
  • 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, 78 pp.
  • S. Hofmann, S. Mayboroda and M. Mourgoglou, Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition, Adv. Math. 270 (2015), 480–564.
  • R. Jiang, J. Xiao and D. Yang, Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants, Anal. Appl. (Singap.) 14 (2016), no. 5, 679–703.
  • T. Ma, P. R. Stinga, J. L. Torrea and C. Zhang, Regularity properties of Schrödinger operators, J. Math. Anal. Appl. 388 (2012), no. 2, 817–837.
  • ––––, Regularity estimates in Hölder spaces for Schrödinger operators via a $T1$ theorem, Ann. Mat. Pura Appl. (4) 193 (2014), no. 2, 561–589.
  • C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166.
  • Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546.
  • L. Song, X. X. Tian and L. X. Yan, On characterization of Poisson integrals of Schrödinger operators with Morrey traces, Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 4, 787–800.
  • E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies 63, Princeton University Press, Princeton, N.J., 1970.
  • E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series 32, Princeton University Press, Princeton, N.J., 1971.
  • L. Tang and J. Han, $L^p$ boundedness for parabolic Schrödinger type operators with certain nonnegative potentials, Forum Math. 23 (2011), no. 1, 161–179.
  • L. Wu and L. Yan, Heat kernels, upper bounds and Hardy spaces associated to the generalized Schrödinger operators, J. Funct. Anal. 270 (2016), no. 10, 3709–3749.
  • M. Yang and C. Zhang, Characterization of temperatures associated to Schrödinger operators with initial data in BMO spaces, arXiv:1710.01160.
  • W. Yuan, W. Sickel and D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010.