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arxiv: 1907.09398 · v1 · pith:FSPEDIOKnew · submitted 2019-07-16 · 🧮 math.CA · math.FA

Morrey spaces for Schr\"odinger operators with certain nonnegative potentials, Littlewood-Paley and Lusin functions on the Heisenberg groups

Hua Wang This is my paper

Pith reviewed 2026-05-24 20:56 UTC · model grok-4.3

classification 🧮 math.CA math.FA
keywords Morrey spacesSchrödinger operatorHeisenberg groupLittlewood-Paley functionLusin area integralreverse Hölder classheat semigroupPoisson semigroup
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The pith

The Littlewood-Paley g-function and Lusin area integral for the Schrödinger operator on the Heisenberg group are bounded on the associated Morrey spaces when the potential lies in the reverse Hölder class RH_q with q at least Q/2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines Morrey spaces tied to the Schrödinger operator L equal to the sublaplacian plus a nonnegative potential V on the Heisenberg group H^n. It then proves that the Littlewood-Paley function g_L and the Lusin area integral S_L, both built from the heat semigroup of L, map these Morrey spaces into themselves. The same conclusion holds for the versions constructed from the Poisson semigroup of the square root of L. The proofs rely on pointwise kernel estimates that follow from the reverse Hölder condition on V. A reader would care because these square functions control regularity and provide substitutes for derivatives in sub-Riemannian harmonic analysis.

Core claim

The author first introduces a class of Morrey spaces associated with L on H^n. Then, using pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of g_L and S_L acting on the Morrey spaces. The same conclusions also hold for the operators g_sqrt(L) and S_sqrt(L) with respect to the Poisson semigroup.

What carries the argument

The Morrey spaces associated with the Schrödinger operator L, together with the pointwise kernel estimates on the derivatives of the heat semigroup that follow from V belonging to RH_q.

If this is right

  • g_L and S_L are bounded operators on the Morrey spaces associated to L.
  • The same boundedness holds when the operators are defined via the Poisson semigroup of sqrt(L).
  • The results apply uniformly for any nonnegative V in RH_q with the stated exponent range on the Heisenberg group of homogeneous dimension Q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The kernel estimates derived from the reverse Hölder condition may transfer other classical boundedness results from the pure sublaplacian case to this perturbed setting.
  • Similar Morrey-space arguments could be tested on other stratified groups where heat-kernel bounds are available.
  • The threshold q equals Q/2 may be sharp; a boundary-case counterexample at that exponent would clarify necessity.

Load-bearing premise

The potential V belongs to the reverse Hölder class RH_q with q at least Q/2, which supplies the pointwise kernel estimates needed for the boundedness proofs.

What would settle it

An explicit potential V in RH_q on H^n together with a function f in the corresponding Morrey space such that the L^p norm of g_L(f) is not controlled by the Morrey norm of f.

read the original abstract

Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_q$ with $q\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. Assume that $\{e^{-s\mathcal L}\}_{s>0}$ is the heat semigroup generated by $\mathcal L$. The Littlewood-Paley function $\mathfrak{g}_{\mathcal L}$ and the Lusin area integral $\mathcal{S}_{\mathcal L}$ associated with the Schr\"odinger operator $\mathcal L$ are defined, respectively, by \begin{equation*} \mathfrak{g}_{\mathcal L}(f)(u) := \bigg(\int_0^{\infty}\bigg|s\frac{d}{ds} e^{-s\mathcal L}f(u) \bigg|^2\frac{ds}{s}\bigg)^{1/2} \end{equation*} and \begin{equation*} \mathcal{S}_{\mathcal L}(f)(u) := \bigg(\iint_{\Gamma(u)} \bigg|s\frac{d}{ds} e^{-s\mathcal L}f(v) \bigg|^2 \frac{dvds}{s^{Q/2+1}}\bigg)^{1/2}, \end{equation*} where \begin{equation*} \Gamma(u) := \big\{(v,s)\in\mathbb H^n\times(0,\infty): |u^{-1}v| < \sqrt{s\,}\big\}. \end{equation*} In this paper the author first introduces a class of Morrey spaces associated with the Schr\"odinger operator $\mathcal L$ on $\mathbb H^n$. Then by using some pointwise estimates of the kernels related to the nonnegative potential $V$, the author establishes the boundedness properties of these two operators $\mathfrak{g}_{\mathcal L}$ and $\mathcal{S}_{\mathcal L}$ acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators $\mathfrak{g}_{\sqrt{\mathcal L}}$ and $\mathcal{S}_{\sqrt{\mathcal L}}$ with respect to the Poisson semigroup $\{e^{-s\sqrt{\mathcal L}}\}_{s>0}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a class of Morrey spaces M^{p,λ}_L associated to the Schrödinger operator L = -Δ_{H^n} + V on the Heisenberg group H^n, where V belongs to the reverse Hölder class RH_q with q ≥ Q/2 and Q = 2n+2 is the homogeneous dimension. It defines the Littlewood-Paley function g_L and Lusin area integral S_L via the heat semigroup e^{-sL}, establishes pointwise kernel estimates for the semigroup and its derivatives from the RH_q assumption, and proves that both g_L and S_L are bounded on M^{p,λ}_L. Analogous boundedness results are claimed for the versions g_{√L} and S_{√L} associated to the Poisson semigroup.

Significance. If the kernel estimates and boundedness proofs hold, the work extends the theory of square functions and area integrals to Morrey spaces in the subelliptic setting with nonnegative potentials satisfying reverse Hölder conditions. This provides a natural adaptation of Euclidean results to stratified groups and could support further analysis of subelliptic Schrödinger equations. The approach credits the standard Calderón-Zygmund techniques adapted to the Heisenberg geometry and the precise range q ≥ Q/2 that yields the required Gaussian upper bounds.

minor comments (3)
  1. [Introduction and main theorems] The ranges of the parameters p and λ for which the Morrey spaces M^{p,λ}_L are defined and the boundedness holds should be stated explicitly in the introduction and in the statement of the main theorems, as these are central to the applicability of the results.
  2. [Section 1 (definitions)] In the definition of the cone Γ(u), the notation |u^{-1}v| should be clarified with respect to the homogeneous norm on H^n to avoid ambiguity for readers unfamiliar with the stratified group structure.
  3. [Introduction] A brief comparison with the corresponding results on Euclidean space (e.g., references to works on Schrödinger operators in R^n) would help situate the contribution, particularly regarding the role of the homogeneous dimension Q.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on Morrey spaces associated to Schrödinger operators on the Heisenberg group, as well as for the recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring point-by-point response at this stage. We are prepared to address any minor issues that may arise during the revision process.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation introduces Morrey spaces M^{p,λ}_L associated to L = -Δ_H^n + V (with V ∈ RH_q, q ≥ Q/2) and then establishes ||g_L f||_{M^{p,λ}_L} ≲ ||f||_{M^{p,λ}_L} (and likewise for S_L) via pointwise heat-kernel bounds that follow from the reverse-Hölder assumption. These kernel estimates are obtained independently of the target boundedness statements and are not defined in terms of the Morrey-space norms or the square-function operators themselves. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain appears; the argument relies on standard Calderón-Zygmund techniques adapted to the Heisenberg geometry and external kernel estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not specify any free parameters, axioms, or invented entities; the work relies on standard assumptions in the field of harmonic analysis on groups.

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    D. R. Adams, Morrey Spaces, Lecture notes in applied and numerical harmonic analysis, Birkh¨ auser/Springer, Cham, 2015

    work page 2015

  2. [2]

    D. R. Adams and J. Xiao, Morrey spaces in harmonic analysis , Ark. Mat, 50(2012), 201–230

    work page 2012

  3. [3]

    D. R. Adams and J. Xiao, Nonlinear potential analysis on Morrey spaces and their capacities , Indiana Univ. Math. J., 53(2004), 1629–1663

    work page 2004

  4. [4]

    Bongioanni, E

    B. Bongioanni, E. Harboure, O. Salinas, Classes of weights related to Schr¨ odinger operators, J. Math. Anal. Appl., 373 (2011), 563–579

    work page 2011

  5. [5]

    Bongioanni, E

    B. Bongioanni, E. Harboure, O. Salinas, Weighted inequalities for com- mutators of Schr¨ odinger-Riesz transforms, J. Math. Anal. Appl., 392 (2012), 6–22

    work page 2012

  6. [6]

    Bongioanni, A

    B. Bongioanni, A. Cabral and E. Harboure, Extrapolation for classes of weights related to a family of operators and applications , Potential Anal., 38 (2013), 1207–1232

    work page 2013

  7. [7]

    Bongioanni, A

    B. Bongioanni, A. Cabral and E. Harboure, Lerner’s inequality associ- ated to a critical radius function and applications , J. Math. Anal. Appl., 407 (2013), 35–55

    work page 2013

  8. [8]

    T. A. Bui, Weighted estimates for commutators of some singular in- tegrals related to Schr¨ odinger operators, Bull. Sci. Math., 138 (2014), 270–292

    work page 2014

  9. [9]

    Dziuba´ nski and J

    J. Dziuba´ nski and J. Zienkiewicz, Hardy spaces associated with some Schr¨ odinger operators, Studia Math, 126 (1997), 149–160

    work page 1997

  10. [10]

    Dziuba´ nski and J

    J. Dziuba´ nski and J. Zienkiewicz, Hardy space H 1 associated to Schr¨ odinger operator with potential satisfying reverse H¨ older inequality, Rev. Mat. Iberoamericana, 15 (1999), 279–296

    work page 1999

  11. [11]

    Dziuba´ nski and J

    J. Dziuba´ nski and J. Zienkiewicz, H p spaces associated with Schr¨ odinger operators with potentials from reverse H¨ older classes, Colloq. Math, 98 (2003), 5–38. 26

    work page 2003

  12. [12]

    Dziuba´ nski,G

    J. Dziuba´ nski,G. Garrig´ os, T. Mart ´ ınez, J. L. Torrea and J.Zienkiewicz, BM O spaces related to Schr¨ odinger operators with potentials satisfying a reverse H¨ older inequality, Math. Z., 249 (2005), 329–356

    work page 2005

  13. [13]

    Di Fazio and M

    G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with d iscontinuous coefficients, J. Funct. Anal, 112(1993), 241–256

    work page 1993

  14. [14]

    Di Fazio, D

    G. Di Fazio, D. K. Palagachev and M. A. Ragusa, Global Morrey regu- larity of strong solutions to the Dirichlet problem for elli ptic equations with discontinuous coefficients , J. Funct. Anal, 166(1999), 179–196

    work page 1999

  15. [15]

    G. B. Folland, Harmonic Analysis in Phase Space , Annals of Mathemat- ics Studies, Princeton Univ. Press, Princeton, New Jersey, 1989

    work page 1989

  16. [16]

    G. B. Folland and E. M. Stein, Estimates for the ¯∂b complex and analysis on the Heisenberg group , Comm. Pure Appl. Math., 27 (1974), 429–522

    work page 1974

  17. [17]

    J. A. Goldstein, Semigroups of Linear Operators and Applications , Ox- ford Univ. Press, New York, 1985

    work page 1985

  18. [18]

    V. S. Guliyev, A. Eroglu and Y. Y. Mammadov, Riesz potential in gen- eralized Morrey spaces on the Heisenberg group , J. Math. Sci. (N.Y.) 189 (2013), 365–382

    work page 2013

  19. [19]

    Hofmann, G

    S. Hofmann, G. Z. Lu, D. Mitrea, M. Mitrea, L. X. Yan, Hardy spaces as- sociated to non-negative self-adjoint operators satisfyi ng Davies-Gaffney estimates, Mem. Amer. Math. Soc., 214 (2011)

    work page 2011

  20. [20]

    R. J. Jiang, X. J. Jiang and D. C. Yang, Maximal function characteriza- tions of Hardy spaces associated with Schr¨ odinger operators on nilpotent Lie groups , Rev. Mat. Complut., 24 (2011), 251–275

    work page 2011

  21. [21]

    H. Q. Li, Estimations Lp des op´ erateurs de Schr¨ odinger sur les groupes nilpotents, J. Funct. Anal., 161 (1999), 152–218

    work page 1999

  22. [22]

    C. C. Lin and H. P. Liu, BM OL(Hn) spaces and Carleson measures for Schr¨ odinger operators, Adv. Math., 228 (2011), 1631–1688

    work page 2011

  23. [23]

    G. Z. Lu, A Fefferman-Phong type inequality for degenerate vector fiel ds and applications , Panamer. Math. J., 6 (1996), 37–57. 27

    work page 1996

  24. [24]

    T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces , Harmonic Analysis, ICM-90 Satellite Conference Pro- ceedings, Springer-Verlag, Tokyo, (1991), 183–189

    work page 1991

  25. [25]

    C. B. Morrey, On the solutions of quasi-linear elliptic partial different ial equations, Trans. Amer. Math. Soc, 43(1938), 126–166

    work page 1938

  26. [26]

    G. X. Pan and L. Tang, Boundedness for some Schr¨ odinger type opera- tors on weighted Morrey spaces , J. Funct. Spaces, 2014, Art. ID 878629, 10 pp

    work page 2014

  27. [27]

    Z. W. Shen, Lp estimates for Schr¨ odinger operators with certain poten- tials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513–546

    work page 1995

  28. [28]

    Song and L

    L. Song and L. X. Yan, Riesz transforms associated to Schr¨ odinger oper- ators on weighted Hardy spaces , J. Funct. Anal, 259 (2010), 1466–1490

    work page 2010

  29. [29]

    E. M. Stein, Singular Integrals and Differentiability Properties of Fun c- tions, Princeton Univ. Press, Princeton, New Jersey, 1970

    work page 1970

  30. [30]

    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality , and Oscillatory Integrals , Princeton Univ. Press, Princeton, New Jersey, 1993

    work page 1993

  31. [31]

    Tang, Weighted norm inequalities for Schr¨ odinger type operators, Fo- rum Math., 27 (2015), 2491–2532

    L. Tang, Weighted norm inequalities for Schr¨ odinger type operators, Fo- rum Math., 27 (2015), 2491–2532

    work page 2015

  32. [32]

    M. E. Taylor, Analysis on Morrey spaces and applications to Navier- Stokes and other evolution equations , Comm. Partial Differential Equa- tions, 17 (1992), 1407–1456

    work page 1992

  33. [33]

    Thangavelu, Harmonic Analysis on the Heisenberg Group , Progress in Mathematics, Vol

    S. Thangavelu, Harmonic Analysis on the Heisenberg Group , Progress in Mathematics, Vol. 159, Birkh¨ auser, Boston/Basel/Berlin, 1998

    work page 1998

  34. [34]

    J. M. Zhao, Littlewood-Paley and Lusin functions on nilpotent Lie groups, Bull. Sci. Math., 132 (2008), 425–438. 28

    work page 2008