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arxiv: 2509.08334 · v2 · pith:YHVWP7SLnew · submitted 2025-09-10 · 🧮 math.FA · math.AP· math.CA

A weak type (p,a) criterion for operators, and applications

Bernhard H. Haak , El-Maati Ouhabaz This is my paper

Pith reviewed 2026-05-18 18:24 UTC · model grok-4.3

classification 🧮 math.FA math.APmath.CA
keywords weak type estimateshomogeneous type spacesRiesz potentialsRiesz transformsspectral multipliersHardy spacesBMO
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The pith

If T is bounded from L^p(Ω) to L^q(Ω) on a space of homogeneous type, then T is of weak type (p0,a) whenever 1/p0 - 1/a equals 1/p - 1/q.

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

The paper gives a criterion that converts the assumption of L^p to L^q boundedness for an operator T into a weak type (p0,a) conclusion under one explicit relation among the four indices. A reader would care because such weak bounds often serve as the missing step when proving mapping properties on other spaces or for concrete integral operators. The criterion is applied to obtain weak estimates for Riesz potentials of the form L to the power minus alpha over 2 and for associated Riesz transforms. It further produces boundedness from the Hardy space H^1_L into L^a and, by duality, from L to the a-prime into the corresponding BMO space.

Core claim

The paper claims that whenever T maps L^p(Ω) boundedly into L^q(Ω) for 1 ≤ p ≤ q < ∞, with Ω open in a space of homogeneous type (X,d,μ), then T is automatically of weak type (p0,a) as soon as the indices satisfy the relation 1/p0 − 1/a = 1/p − 1/q. This abstract statement is illustrated by deriving the corresponding weak bounds for the Riesz potential L^{−α/2} and the operator ∇Δ^{−α/2}, and by showing L^p−L^q boundedness for spectral multipliers F(L) when the heat kernel of L obeys a Gaussian upper bound or an off-diagonal estimate. The same results also give boundedness from the Hardy space H^1_L associated with L into L^a(X), and by duality from L^{a′}(X) into BMO_L.

What carries the argument

The weak-type (p0,a) criterion that follows directly from L^p-to-L^q boundedness once the reciprocal-difference relation 1/p0 − 1/a = 1/p − 1/q is imposed.

If this is right

  • Weak type (p0,a) estimates hold for the Riesz potential L^{−α/2}.
  • The same weak estimates apply to Riesz-transform-type operators of the form ∇Δ^{−α/2}.
  • L^p to L^q boundedness holds for spectral multipliers F(L) whenever the heat kernel of L satisfies Gaussian upper bounds or off-diagonal estimates.
  • The operators map the associated Hardy space H^1_L boundedly into L^a(X).
  • By duality the operators map L^{a′}(X) boundedly into BMO_L.

Where Pith is reading between the lines

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

  • The same relation between indices may produce weak-type conclusions for operators defined by off-diagonal kernel bounds without first establishing full L^p to L^q boundedness.
  • The criterion could be tested on doubling metric measure spaces that are not necessarily spaces of homogeneous type in the classical sense.
  • Endpoint mapping properties obtained this way may combine with existing Calderón-Zygmund theory to reach other function spaces such as Lorentz or Orlicz classes.

Load-bearing premise

The operator T must be bounded from L^p(Ω) to L^q(Ω) on an open subset Ω of a space of homogeneous type.

What would settle it

An explicit operator that remains bounded from L^p(Ω) to L^q(Ω) but fails the weak (p0,a) bound when the underlying space is not of homogeneous type would disprove the criterion.

read the original abstract

Let $(X, d, \mu)$ be a space of homogeneous type and $\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\Omega) \to L^q(\Omega)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\frac{1}{p_0} - \frac{1}{a} = \frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\frac{\alpha}{2}}$ or for Riesz transform type operators $\nabla \Delta^{-\frac{\alpha}{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H^1_L$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a'}(X)$ into $\text{BMO}_L$.

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

2 major / 2 minor

Summary. The paper establishes a criterion on spaces of homogeneous type (X,d,μ) with Ω open: if T is bounded L^p(Ω)→L^q(Ω) for 1≤p≤q<∞, then T satisfies a weak-type (p0,a) bound whenever 1/p0−1/a=1/p−1/q. The criterion is applied to obtain weak-type estimates for Riesz potentials L^{−α/2} and Riesz transforms ∇Δ^{−α/2}, L^p-L^q bounds for spectral multipliers F(L) under Gaussian or off-diagonal heat-kernel assumptions, and boundedness of these operators from the associated Hardy space H^1_L into L^a(X), with duality yielding L^{a'}→BMO_L.

Significance. If the central criterion holds, the work supplies a flexible tool for converting strong-type bounds into weak-type and endpoint estimates on doubling metric measure spaces, a setting that appears frequently in harmonic analysis. The applications to Riesz potentials, spectral multipliers, and Hardy-space mappings are concrete and potentially reusable; the manuscript also ships explicit applications to operators whose kernels satisfy standard Gaussian bounds.

major comments (2)
  1. [§2, Theorem 2.1] §2, Theorem 2.1: the weak-type criterion is stated in terms of a level-set estimate involving the doubling constant; it is not immediately clear whether the constant depends on the choice of p0 and a or remains uniform when the exponent relation is fixed. A short remark clarifying the dependence would strengthen the statement.
  2. [§4.2] §4.2, application to Riesz potentials: the passage from the L^p→L^q bound to the weak (p0,a) bound for L^{−α/2} invokes the criterion directly, but the verification that the operator satisfies the required level-set hypothesis is only sketched. An explicit reference to the covering lemma used would make the argument self-contained.
minor comments (2)
  1. Notation: the symbol a is used both for the weak-type second index and, later, for the conjugate exponent in the duality statement; a brief clarification or change of symbol would avoid confusion.
  2. References: the introduction cites several works on spaces of homogeneous type but omits a direct pointer to the original Coifman–Weiss monograph; adding it would help readers locate the doubling-property background.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for the constructive comments. We address each of the major comments below and have incorporated revisions to enhance the clarity of the paper.

read point-by-point responses

  1. Referee: [§2, Theorem 2.1] §2, Theorem 2.1: the weak-type criterion is stated in terms of a level-set estimate involving the doubling constant; it is not immediately clear whether the constant depends on the choice of p0 and a or remains uniform when the exponent relation is fixed. A short remark clarifying the dependence would strengthen the statement.

    Authors: We are grateful for this suggestion. The constant appearing in the weak-type (p0, a) estimate of Theorem 2.1 depends on the doubling constant of the space, the fixed exponents p and q, the choice of p0 and a satisfying the given relation, and the operator norm of T. However, for a fixed relation 1/p0 − 1/a = 1/p − 1/q, the dependence is uniform in the sense that it does not vary with other parameters when they are held fixed. We have added a short remark following Theorem 2.1 in the revised manuscript to clarify this dependence explicitly. revision: yes

  2. Referee: [§4.2] §4.2, application to Riesz potentials: the passage from the L^p→L^q bound to the weak (p0,a) bound for L^{−α/2} invokes the criterion directly, but the verification that the operator satisfies the required level-set hypothesis is only sketched. An explicit reference to the covering lemma used would make the argument self-contained.

    Authors: We thank the referee for pointing this out. To make the argument self-contained, we have revised the text in §4.2 to include an explicit citation to the covering lemma employed in the verification of the level-set hypothesis for the Riesz potential. This addition clarifies the steps without changing the overall approach. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central result is a criterion that takes the given L^p(Ω)→L^q(Ω) boundedness of T together with the doubling property of the homogeneous space and standard covering arguments to conclude weak-type (p0,a) boundedness under the stated exponent relation. This relation is an algebraic identity on the indices and is not derived from the conclusion; the proof supplies an independent level-set or distribution-function estimate that is not presupposed by the input boundedness. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain. The applications to Riesz potentials and spectral multipliers are presented as illustrations that invoke the criterion rather than circularly justify it.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the standard definition of spaces of homogeneous type and the hypothesis of L^p-L^q boundedness; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • domain assumption The space (X, d, μ) is of homogeneous type and Ω is an open subset of X.
    Explicitly stated as the setting in the first sentence of the abstract.
  • domain assumption T is a bounded operator from L^p(Ω) to L^q(Ω) for some 1 ≤ p ≤ q < ∞.
    Given as the hypothesis from which the weak-type conclusion is derived.

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Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Auscher,On necessary and sufficient conditions forL p-estimates of riesz transforms associated to elliptic operators onR n and related estimates, Mem

    P. Auscher,On necessary and sufficient conditions forL p-estimates of riesz transforms associated to elliptic operators onR n and related estimates, Mem. Am. Math. Soc., vol. 871, Providence, RI: American Mathematical Society (AMS), 2007 (English)

    work page 2007

  2. [2]

    M. T. Barlow and R. F. Bass,Brownian motion and harmonic analysis on Sierpinski carpets, Can. J. Math.51(1999), no. 4, 673–744

    work page 1999

  3. [3]

    Blunck and P

    S. Blunck and P. C. Kunstmann,Calder´ on-Zygmund theory for non-integral operators and theH ∞ functional calculus, Rev. Mat. Iberoam.19(2003), no. 3, 919–942

    work page 2003

  4. [4]

    Coulhon,Dimension at infinity of an analytic semigroup., Bull

    T. Coulhon,Dimension at infinity of an analytic semigroup., Bull. Sci. Math., II. S´ er.114(1990), no. 4, 485–500 (French)

    work page 1990

  5. [5]

    Coulhon and X

    T. Coulhon and X. T. Duong,Riesz transforms for1≤p≤2, Trans. Am. Math. Soc.351(1999), no. 3, 1151–1169

    work page 1999

  6. [6]

    E. B. Davies,Heat kernels and spectral theory, Camb. Tracts Math., vol. 92, Cambridge etc.: Cam- bridge University Press, 1989

    work page 1989

  7. [7]

    X. T. Duong and A. McIntosh,Singular integral operators with non-smooth kernels on irregular do- mains, Rev. Mat. Iberoam.15(1999), no. 2, 233–265

    work page 1999

  8. [8]

    X. T. Duong, E. M. Ouhabaz, and A. Sikora,Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal.196(2002), no. 2, 443–485

    work page 2002

  9. [9]

    X. T. Duong, E. M. Ouhabaz, and L. Yan,Endpoint estimates for Riesz transforms of magnetic Schr¨ odinger operators, Ark. Mat.44(2006), no. 2, 261–275

    work page 2006

  10. [10]

    X. T. Duong and L. Yan,Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Am. Math. Soc.18(2005), no. 4, 943–973

    work page 2005

  11. [11]

    Dziuba´ nski, G

    J. Dziuba´ nski, G. Garrig´ os, T. Mart´ ınez, J. L. Torrea, and J. Zienkiewicz,BMO spaces related to Schr¨ odinger operators with potentials satisfying a reverse H¨ older inequality, Math. Z.249(2005), no. 2, 329–356

    work page 2005

  12. [12]

    Dziuba´ nski and J

    J. Dziuba´ nski and J. Zienkiewicz,Hardy spaceH 1 associated to Schr¨ odinger operator with potential satisfying reverse H¨ older inequality, Rev. Mat. Iberoam.15(1999), no. 2, 279–296 (English)

    work page 1999

  13. [13]

    A. F. M. ter Elst and E.-M. Ouhabaz,Partial spectral multipliers and partial Riesz transforms for degenerate operators, Rev. Mat. Iberoam.29(2013), no. 2, 691–713

    work page 2013

  14. [14]

    Garc´ ıa-Cuerva and A

    J. Garc´ ıa-Cuerva and A. E. Gatto,Boundedness properties of fractional integral operators associated to non-doubling measures, Stud. Math.162(2004), no. 3, 245–261

    work page 2004

  15. [15]

    Grafakos,Classical and modern Fourier analysis, Upper Saddle River, NJ : Pearson/Prentice Hall, 2004

    L. Grafakos,Classical and modern Fourier analysis, Upper Saddle River, NJ : Pearson/Prentice Hall, 2004

    work page 2004

  16. [16]

    Grafakos, L

    L. Grafakos, L. Liu, and D. Yang,Vector-valued singular integrals and maximal functions on spaces of homogeneous type, Math. Scand.104(2009), no. 2, 296–310

    work page 2009

  17. [17]

    Grigor’yan,Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J

    A. Grigor’yan,Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal.127(1995), no. 2, 363–389

    work page 1995

  18. [18]

    ,Gaussian upper bounds for the heat kernel on arbitrary Riemannian manifolds, J. Differ. Geom.45(1997), no. 1, 33–52

    work page 1997

  19. [19]

    Grigor’yan, J

    A. Grigor’yan, J. Hu, and K.-S. Lau,Heat kernels on metric spaces with doubling measure, Fractal geometry and stochastics IV. Proceedings of the 4th conference, Greifswald, Germany, September 8–12, 2008, Basel: Birkh¨ auser, 2009, pp. 3–44

    work page 2008

  20. [20]

    Hofmann, G

    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. Am. Math. Soc., vol. 1007, Providence, RI: American Mathematical Society (AMS), 2011 (English)

    work page 2011

  21. [21]

    H¨ ormander,Estimates for translation invariant operators inL p spaces, Acta Math.104(1960), 93–140

    L. H¨ ormander,Estimates for translation invariant operators inL p spaces, Acta Math.104(1960), 93–140

    work page 1960

  22. [22]

    Rozendaal and M

    J. Rozendaal and M. Veraar,Fourier multiplier theorems involving type and cotype, J. Fourier Anal. Appl.24(2018), no. 2, 583–619. A WEAK TYPE (p, a) CRITERION FOR OPERATORS, AND APPLICATIONS 25

    work page 2018

  23. [23]

    E. M. Stein,Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. No. 30, Princeton University Press, Princeton, NJ, 1970

    work page 1970

  24. [24]

    M. H. Taibleson and G. Weiss,The molecular characterization of certain Hardy spaces, Representation theorems for Hardy spaces, Paris: Soci´ et´ e Math´ ematique de France (SMF), 1980, pp. 67–151 (English). Institut de Math´ematiques de Bordeaux, Universit´e de Bordeaux, UMR CNRS 5251, 351 Cours de la Lib ´eration 33405, Talence. France Email address:Bernh...

    work page 1980