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arxiv: 2605.18235 · v1 · pith:CZ7RRD3Xnew · submitted 2026-05-18 · 🧮 math.AP

Gradient estimates for singular elliptic measure data problems with double phase

Kyeong Song , Yeonghun Youn , Anna Zatorska-Goldstein This is my paper

Pith reviewed 2026-05-20 09:08 UTC · model grok-4.3

classification 🧮 math.AP MSC 35J6035B65
keywords double phasemeasure dataCalderón-Zygmund estimatessingular elliptic equationsgradient estimatesnonlinear PDEvariable growth
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The pith

Local Calderón-Zygmund estimates are established for singular double-phase elliptic problems with measure data when 2-1/n < p < 2

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

The paper examines divergence-form elliptic equations that combine two different growth rates through the vector field |Du|^{p-2}Du + a(x)|Du|^{q-2}Du with p less than q and a nonnegative coefficient a. The right-hand side is allowed to be a general measure μ. The central result establishes local Calderón-Zygmund estimates controlling the gradient in the singular range 2 minus 1 over n less than p less than 2. These bounds matter because they convert information about the measure into higher integrability of the gradient, which in turn supports further regularity analysis for solutions. The proof operates under natural assumptions on the exponents, the coefficient a, and the measure μ.

Core claim

We prove local Calderón–Zygmund estimates in the singular case 2-1/n < p < 2 for solutions of the double-phase equation -div(|Du|^{p-2}Du + a(x)|Du|^{q-2}Du) = μ in a bounded domain of R^n under natural assumptions on p, q and a(·).

What carries the argument

The double-phase vector field |ξ|^{p-2}ξ + a(x)|ξ|^{q-2}ξ together with local testing and covering arguments that yield integrability upgrades for the gradient in the presence of measure data.

If this is right

  • The gradient Du gains higher local integrability controlled by the data μ.
  • The estimates are local and therefore apply directly to interior regularity questions.
  • The result extends Calderón-Zygmund theory from the regular case p ≥ 2 into the singular regime.
  • The bounds remain uniform under the stated structural conditions on a(x).

Where Pith is reading between the lines

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

  • The same testing strategy may adapt to parabolic double-phase equations with measure data.
  • Boundary regularity versions could follow by combining the interior estimates with suitable boundary assumptions.
  • The capacity conditions on μ may link to removability criteria for singularities in related double-phase problems.

Load-bearing premise

Natural assumptions on the coefficient a(·), the relation between p and q, and the integrability or capacity properties of the measure μ suffice to close the estimates.

What would settle it

A concrete counterexample consisting of a measure μ, coefficient a(x), and exponents satisfying the natural assumptions for which the local gradient integrability bound fails in the range 2-1/n < p < 2 would disprove the claim.

read the original abstract

We consider elliptic measure data problems of the type \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu \] in a bounded domain in $\mathbb{R}^n$, where $p<q$ and $a(\cdot) \ge 0$. We prove local Calder\'on--Zygmund estimates in the singular case $2-1/n < p < 2$, under natural assumptions on $p$, $q$ and $a(\cdot)$.

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

1 major / 2 minor

Summary. The paper considers elliptic measure data problems of the form -div(|Du|^{p-2}Du + a(x)|Du|^{q-2}Du) = μ in a bounded domain in R^n, with p < q and a(·) ≥ 0. It proves local Calderón-Zygmund estimates in the singular case 2-1/n < p < 2, under natural assumptions on p, q and a(·).

Significance. If the result holds, it provides important gradient integrability estimates for degenerate elliptic operators with double phase and measure data in the singular regime, extending existing theory and potentially aiding in the study of regularity for such PDEs. The manuscript includes a detailed proof, which is a positive aspect.

major comments (1)
  1. The 'natural assumptions' on a(·), the relation between p and q, and the properties of μ are invoked throughout the estimates. It would be helpful to explicitly state in §1 or §2 whether these include Hölder continuity of a(·) and μ in the dual of W^{1,p}, and confirm that they suffice to absorb the q-term near p = 2 - 1/n.
minor comments (2)
  1. The abstract could specify the precise assumptions on a(·) to make the claim more self-contained.
  2. Ensure consistent use of notation for the double phase operator throughout the manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The suggestion to clarify the natural assumptions improves the presentation, and we have incorporated it as a minor revision.

read point-by-point responses

  1. Referee: The 'natural assumptions' on a(·), the relation between p and q, and the properties of μ are invoked throughout the estimates. It would be helpful to explicitly state in §1 or §2 whether these include Hölder continuity of a(·) and μ in the dual of W^{1,p}, and confirm that they suffice to absorb the q-term near p = 2 - 1/n.

    Authors: We appreciate the referee's suggestion for greater explicitness. The assumptions on a(·) (Hölder continuity with exponent α > 0) and on μ (membership in the dual of W^{1,p}_0(Ω)) are stated in Section 2 and used throughout the proofs. These conditions are sufficient to absorb the q-term near the lower threshold p = 2 - 1/n, as the Hölder regularity of a(·) controls the perturbation and the dual-space integrability of μ ensures the singular estimates remain valid. In the revised manuscript we have added a dedicated paragraph in §1.2 that lists the assumptions verbatim and includes a short remark confirming the absorption step. This is a clarification only and does not alter any proofs or results. revision: yes

Circularity Check

0 steps flagged

No circularity detected; standard PDE proof with independent derivation chain

full rationale

This is a theoretical mathematics paper establishing local Calderón-Zygmund gradient estimates via comparison principles, Gehring-type lemmas, and capacity estimates for a double-phase elliptic operator with measure data. The derivation relies on explicit assumptions on the coefficient a(·), the relation p < q, and integrability properties of μ, without any reduction of the central claim to fitted parameters, self-definitional steps, or load-bearing self-citations. All steps in the proof chain are constructed from standard analytic tools external to the result itself, rendering the argument self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results from elliptic regularity theory, Sobolev spaces, and measure theory that are not introduced in the abstract.

axioms (1)
  • standard math Standard structural assumptions on the double-phase operator and suitable integrability or capacity conditions on the measure μ.
    Invoked implicitly to obtain the stated estimates.

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Reference graph

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