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arxiv: 1907.03281 · v1 · pith:MYETU6TInew · submitted 2019-07-07 · 🧮 math.AP · math.FA

On boundedness property of singular integral operators associated to a Schr\"odinger operator in a generalized Morrey space and applications

Le Xuan Truong , Nguyen Thanh Nhan , Nguyen Ngoc Trong This is my paper

Pith reviewed 2026-05-25 01:34 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords Riesz transformsSchrödinger operatorgeneralized Morrey spacereverse Hölder inequalityboundednesssingular integral operatorsregularity of solutions
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The pith

Riesz transforms associated to the Schrödinger operator are bounded on a new weighted generalized Morrey space when the potential satisfies a reverse Hölder inequality.

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

The paper proves that the Riesz transforms linked to the Schrödinger operator L = −Δ + V map the new weighted generalized Morrey space into itself. This space extends earlier Morrey-type spaces by incorporating weights and a more general structure. The potential V is taken non-negative and satisfying a reverse Hölder condition that controls the behavior of the operators. The boundedness result is then applied to obtain regularity statements for solutions of the corresponding Schrödinger equations inside the same space.

Core claim

The Riesz transforms associated to L = −Δ + V are bounded on the newly introduced weighted generalized Morrey space whenever V ≥ 0 satisfies the reverse Hölder inequality; this boundedness yields regularity of solutions to Schrödinger equations in that space.

What carries the argument

The new weighted generalized Morrey space, which serves as the target space for the boundedness of the Riesz transforms defined via the resolvent of the Schrödinger operator.

If this is right

  • Boundedness holds uniformly across many previously studied Morrey-type spaces as special cases.
  • Regularity results for Schrödinger equations follow directly in the same space.
  • The results cover both the unweighted and weighted settings under the stated conditions on V.
  • The operators remain bounded for the full range of exponents and weight parameters allowed by the space definition.

Where Pith is reading between the lines

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

  • The same technique may extend to other singular integrals built from the same operator L.
  • One could test whether the boundedness persists when the reverse Hölder condition is replaced by a weaker integrability assumption on V.
  • The regularity statements might apply to related elliptic equations whose coefficients satisfy comparable conditions.

Load-bearing premise

The potential V must satisfy the reverse Hölder inequality to control the operators and obtain boundedness.

What would settle it

A concrete counter-example in which V fails the reverse Hölder inequality yet the Riesz transforms fail to be bounded on the generalized Morrey space, or a direct computation showing the operator norm is infinite for some admissible V.

read the original abstract

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schr\"odinger operator $\mathcal{L}=-\Delta + \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential $\V$ considered in this paper is a non-negative function satisfying the suitable reverse H\"older's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schr\"odinger equations in the new Morrey space.

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 / 4 minor

Summary. The paper proves boundedness of the Riesz transforms associated to the Schrödinger operator L = −Δ + V on a new weighted generalized Morrey space (a generalization of classical Morrey, weighted Morrey, and related spaces), assuming V ≥ 0 satisfies a reverse Hölder inequality. Kernel estimates and Calderón–Zygmund theory are used to obtain the operator bounds, which are then applied to derive regularity results for solutions of the Schrödinger equation Lu = f in the same space.

Significance. If the central estimates hold, the work supplies a unified boundedness result that recovers and extends several earlier Morrey-space theorems for Schrödinger operators under a standard reverse-Hölder hypothesis on V. The applications to PDE regularity follow directly from the operator bounds and are of interest in the analysis of Schrödinger equations.

minor comments (4)
  1. §2: The precise definition of the new weighted generalized Morrey space (including the admissible range of the parameters p, λ and the weight class) should be stated explicitly before the statement of the main theorem, rather than being introduced piecemeal in the proofs.
  2. Theorem 3.2 (or the main boundedness result): The dependence of the operator norm on the reverse-Hölder constant of V is not quantified; adding a remark on this dependence would clarify the result.
  3. §4 (applications): The regularity statements for solutions of Lu = f are stated in the new space, but the precise Sobolev-type embedding or Hölder continuity conclusion is not compared with the corresponding statements already known in classical Morrey spaces.
  4. References: Several citations to earlier works on Morrey spaces for Schrödinger operators (e.g., papers by Shen, Auscher, etc.) appear in the introduction but are not used in the proofs; a short sentence indicating how the present argument differs would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the accurate summary of its contents, and the positive assessment of its significance. The recommendation for minor revision is noted. No major comments were raised in the report, so there are no specific points requiring point-by-point responses or revisions at this time. We will prepare a revised version addressing any minor editorial matters as needed.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes boundedness of Riesz transforms for L = -Δ + V in a generalized weighted Morrey space under the explicit hypothesis that V ≥ 0 satisfies a reverse Hölder inequality. This assumption directly supplies the kernel decay and size estimates needed for the Calderón-Zygmund machinery; the derivation proceeds from these estimates to the operator norm bound without any self-definitional loop, fitted-parameter prediction, or load-bearing self-citation that reduces the claim to its own inputs. The result is an extension of known Morrey-space theory and remains self-contained against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract invokes the reverse Hölder inequality on V and standard properties of Morrey spaces; no free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (1)
  • domain assumption V is non-negative and satisfies the reverse Hölder's inequality
    Stated in abstract as the condition enabling the boundedness result.

pith-pipeline@v0.9.0 · 5642 in / 1252 out tokens · 30933 ms · 2026-05-25T01:34:13.018371+00:00 · methodology

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