A $T1$ criterion for Schrödinger-Calderón-Zygmund operators with exponential decay

Abstract: We establish the boundedness of exponential Schrödinger-Calderón-Zygmund operators on weighted $BMO_ρ<sup>α(w)$ spaces via a $T1$ criterion, where the weights belong to classes that capture the exponential decay of the operators, and $ρ$ is a critical radius function. Specifically, we prove that the boundedness of such an operator $T$ on $BMO_ρ<sup>α(w)$ is equivalent to a natural oscillation condition on $T1$ over sub-critical balls. The weight classes considered, introduced in connection with $ρ$, include and extend the classical $A_p<sup>ρ$ weights, and are well-adapted to the exponential decay of the kernels. As applications, we derive weighted endpoint estimates for several operators associated to the generalized Schrödinger operator $\mathcal{L}_μ=-Δ+μ$, including Riesz transforms, Laplace transform-type multipliers, maximal operators for the heat and Poisson semigroups, Littlewood-Paley functions and fractional integral operators. When $dμ(x)=V(x)dx$, the results above extend the known endpoint estimates to larger classes of weights.

• The paper establishes a T1-type criterion for the boundedness of exponential SCZO operators via precise T1 oscillation estimates on weighted BMO spaces.
• It develops a framework involving critical radius functions and exponential decay to generalize classical Calderón-Zygmund theory with sharp kernel estimates.
• The findings yield endpoint and weighted boundedness results for generalized Riesz transforms, Laplace multipliers, and other operators in Schrödinger-type settings.

$T1$ Criterion for Schrödinger-Calderón-Zygmund Operators with Exponential Decay

Overview and Motivation

The article "A $T1$ criterion for Schrödinger-Calderón-Zygmund operators with exponential decay" (2607.02705) establishes a $T1$-type characterization for the boundedness of a broad class of Schrödinger-Calderón-Zygmund operators (SCZOs) exhibiting exponential decay of their kernels, extending the Calderón-Zygmund theory to a context governed by a critical radius function $\rho$ that encodes the geometry associated with a generalized Schrödinger operator $\mathcal{L}_\mu = -\Delta + \mu$, where $\mu$ is a non-negative Radon measure or a potential.

The main result is a $T1$ criterion (analogous to the classical David-Journé theorem) for the boundedness of exponential-decay SCZOs (and their fractional analogues) on weighted $BMO^\alpha_\rho(w)$ spaces, with weight classes adapted to the exponential decay and the underlying geometry. The analysis systematically generalizes classical results by incorporating nonpolynomial (exponential) decay, providing precise endpoint, weighted, and fractional estimates.

Technical Framework

Critical Radius Function and Adapted Geometry

A critical radius function $\rho:\mathbb{R}^d \to [0,\infty)$ measures the local geometric influence of the potential or measure $\mu$ in Schrödinger-type settings. It satisfies two-sided polynomial comparability with respect to the normalized distance $T1$0, ensuring near-constant behavior on local (sub-critical) scales and controlled variation on larger scales.

Sub-critical and super-critical balls are defined using $T1$1, and the family of sub-critical balls $T1$2 forms the geometric building block for associated function spaces and weights.

Weighted $T1$3 Spaces

For $T1$4 and a weight $T1$5, the space $T1$6 consists of functions with mean oscillation controlled on sub-critical balls by $T1$7 and for super-critical balls by a similar $T1$8-type average. The norm is suitably defined to capture this dual local/global regularity, and these spaces generalize both unweighted BMO and H\"older-type spaces depending on $T1$9.

Exponential Schrödinger-Calderón-Zygmund Operators

An operator $T1$0 is an exponential SCZO of type $T1$1 (with $T1$2) if it is of weak type $T1$3 and has an associated kernel $T1$4 satisfying:

• Size: For $T1$5,

$T1$6

• Smoothness:

$T1$7

For $T1$8, these are replaced by pointwise estimates. These conditions generalize the classical Calderón-Zygmund theory to the exponentially decaying setting imposed by Schrödinger-type operators.

Exponentially Adapted Weight Classes

The analysis introduces and employs weight classes $T1$9 and their intersections with reverse Hölder and doubling classes, $\rho$0 and $\rho$1. These weights are well-suited to the exponential decay, exceed the classical $\rho$2 classes, and are defined by exponential-type bounds on weighted averages.

Main Results

The $\rho$3 Criterion

Let $\rho$4 be an exponential SCZO of type $\rho$5 with parameters $\rho$6. For $\rho$7 and $\rho$8, the following are equivalent:

1. Oscillation Condition: For all $\rho$9 with $\mathcal{L}_\mu = -\Delta + \mu$0,

$\mathcal{L}_\mu = -\Delta + \mu$1

(or a logarithmic bound if $\mathcal{L}_\mu = -\Delta + \mu$2 and $\mathcal{L}_\mu = -\Delta + \mu$3).

1. Weighted BMO Boundedness: $\mathcal{L}_\mu = -\Delta + \mu$4 is bounded for weights $\mathcal{L}_\mu = -\Delta + \mu$5 with parameter restrictions explicitly tracking the exponential decay and weight growth, i.e., $\mathcal{L}_\mu = -\Delta + \mu$6 must be small relative to the decay.
2. Boundedness for Power Weights: $\mathcal{L}_\mu = -\Delta + \mu$7 is bounded when $\mathcal{L}_\mu = -\Delta + \mu$8 with uniform operator norm in $\mathcal{L}_\mu = -\Delta + \mu$9.

This characterization mirrors, but substantially generalizes, the classical $\mu$0 theorem: boundedness is determined by the action of the operator on constants via an explicit oscillation estimate for $\mu$1. The result covers both polynomial and exponential decay (the latter being essential for Schrödinger-type heat, Poisson, and Riesz transforms with nontrivial potentials or measures).

Weighted and Endpoint Applications

Applications of the criterion are developed for a wide array of operators:

• Generalized Riesz transforms $\mu$2, $\mu$3 (and their fractional/smooth analogues), with optimal boundedness results across $\mu$4 and $\mu$5 for large classes of exponential-type weights and endpoint spaces.
• Laplace transform-type multipliers $\mu$6, maximal heat and Poisson semigroup operators, and Littlewood–Paley-type square functions—all shown to satisfy the $\mu$7 criterion and thus admit endpoint, weighted boundedness.
• Fractional integrals $\mu$8 and mixed operators $\mu$9, with explicit dependence of operator norm and domain/range spaces on $T1$0, the geometry, and the underlying weighting.

Strong, explicit endpoint bounds are provided for these classes, notably with exponential weights that were not accessible via previous polynomial techniques. For many operators, this extends and refines classical results for Schrödinger operators with $T1$1 weights, including in contexts with nonstandard measures and unbounded potentials.

Methodology and Novel Contributions

The technical core is an overview of:

• Detailed kernel estimates for SCZOs in the presence of exponentially decaying geometry, adapted from sharp heat and Green function bounds for general Schrödinger operators with measure or potential.
• Construction and analysis of exponentially adapted weight classes, carefully controlling their growth and compatibility with kernel estimates on all scales, via critical radius and Agmon distance techniques.
• Development of vector-valued, endpoint, and unweighted/weighted boundedness regimes, including operator families not previously covered by polynomial-weighted theory.
• Deployment of refinement of oscillation criteria for $T1$2 in the exponentially decaying setting, including precise treatment of the logarithmic endpoint case.

The equivalence result is robust: failure of the oscillation estimate on sub-critical balls always witnesses failure of boundedness in the corresponding weighted BMO setting. The theory operates at a sharp, nearly optimal level across the class of SCZOs considered.

Implications and Future Directions

The $T1$3 criterion for exponential SCZOs facilitates the systematic extension of harmonic analysis and PDE theory for Schrödinger-type operators to exponentially weighted and endpoint settings, advancing beyond traditional polynomial frameworks. Practical implications include:

• PDE/Evolution equations: The results provide endpoint and weighted control for heat, Poisson, and functional calculus solutions under potentials and singular measures, relevant in quantum mechanics, dispersive PDE, and stochastic processes.
• Harmonic analysis: The methods extend boundedness theory for singular integral, maximal, and square function operators to geometries dictated by measure and potential, suggesting further development for multilinear settings and critical-exponent phenomena.
• Weighted theory: The framework paves the way for further study on sharp constants, optimal growth ranges, and extrapolation in non-polynomial-weighted regimes—potentially informing applications in regularity and degenerate elliptic theory.

Open directions include systematic treatment of commutators, variable coefficient generalizations, extensions to more general geometric settings (e.g., metric measure spaces or manifolds), and nonlinear or vector-valued generalizations—many of which can leverage the explicit critical radius and oscillation formalism developed.

Conclusion

The article provides a comprehensive and technically deep extension of the $T1$4 theorem to the class of exponential-decay Schrödinger-Calderón-Zygmund and fractional operators, completely characterizing their endpoint and weighted boundedness on spaces $T1$5 with respect to exponential-type weights. The analysis is sharp both in the oscillation criteria and in the weight classes involved, with broad applicability to the advanced harmonic analysis of Schrödinger-type operators. The established framework offers a precise analytic toolkit for further advances in PDEs, potential theory, and weighted singular integral analysis in complex or non-smooth environments.