Unboundedness of Riesz Operator

• Unboundedness of the Riesz operator refers to the failure of this singular integral operator to map L^p spaces boundedly, especially in non-doubling and fractal geometries.
• This phenomenon is driven by geometric constraints such as dimension mismatches on manifolds with ends and the breakdown of Sobolev–Poincaré inequalities.
• Rigorous analysis using spectral splitting, explicit counterexamples, and Hardy–Hilbert inequalities provides sharp thresholds for the operator’s boundedness.

The Riesz operator is a fundamental object in harmonic analysis and geometric analysis, acting as a first-order singular integral operator deeply connected with the geometry and spectral properties of the underlying space. Its boundedness properties on $L^p$ spaces are both classical in Euclidean space and highly subtle in noncompact, inhomogeneous, or fractal settings. Unboundedness of the Riesz operator—failure of this operator to map $L^p$ into $L^p$ boundedly—serves as a diagnostic of intricate structural limitations: non-doubling geometry, dimension mismatch on manifolds with ends, failures of Sobolev and Poincaré inequalities, and the presence of irregular or fractal measures. This article systematically surveys the mechanisms and phenomena under which the Riesz operator loses its $L^p$ boundedness, referencing rigorous results across a variety of geometric and analytic contexts.

1. Riesz Operator: Definition and Contexts

On a Riemannian manifold $(\mathcal{M},g)$, the Riesz operator is defined as $R = \nabla \Delta^{-1/2}$, where $\Delta$ is the Laplace–Beltrami operator and $\nabla$ the Riemannian gradient. In Euclidean space, $R_j$ is given either by convolution with the kernel $K_j(x) = c_n \frac{x_j}{|x|^{n+1}}$ or as the Fourier multiplier $i \xi_j / |\xi|$. In general, the operator is interpreted via functional calculus as

$$R f = \frac{2}{\pi} \int_{0}^{\infty} \nabla(\Delta + k^2)^{-1} f \, dk ,$$

admitting spectral splitting into low- and high-energy contributions. For manifolds with ends of different asymptotic dimensions, the operator is defined analogously, with the noncompact geometry dictating the decay and integrability of the associated kernel (Hassell et al., 2018).

2. Unboundedness Mechanisms on Manifolds with Ends

Consider manifolds of the form

$$\mathcal{M} = (\mathbb{R}^{n_1} \times M_1) \# \cdots \# (\mathbb{R}^{n_l} \times M_l),$$

where each $M_i$ is compact and the ends may have differing Euclidean dimensions $n_i \geq 3$. The operator $R$ exhibits $L^p$ boundedness only for $1 < p < n_*$, with $n_* = \min_i n_i$; it fails for $p \ge n_*$ (He, 2024, Hassell et al., 2018). The low-energy resolvent kernel on the $i$-th end decays as $|z'-z|^{1-n_i}$, and this is integrable only for $p < n_i$. The singular behavior is sharply localized in the parametrix construction: for $p$ at or above the minimal Euclidean dimension, rank-one contributions with slow decay dominate, and classical Hardy–Hilbert inequalities confirm the exact sharpness of the threshold (He, 2023).

3. Endpoint and Lorentz Space Phenomena

When $p$ equals $n_* = \min_i n_i$, even Lorentz spaces fail to retain boundedness: there is no $L^{n_*,q}$ bound for $q>1$ (He, 2023). The operator sends finite norm inputs at this endpoint to infinite norm outputs via explicit radial counterexamples. The underlying convolution structure associated with $|z'-z|^{1-n_*}$ and volume measure $|z'-z|^{n_*-1} d|z'-z|$ aligns exactly with the breakdown of Hardy–Hilbert inequalities in the critical case.

4. Unboundedness Due to Geometry, Doubling Failures, and Exterior Domains

Non-doubling manifolds—those for which volume growth of balls does not satisfy uniform scaling—lead inexorably to unboundedness of Riesz transforms at specific exponents. Similarly, in exterior Lipschitz domains, the Riesz transform fails to be $L^p$-bounded for $p \ge n$ (with $n$ the ambient Euclidean dimension), due to the presence of harmonic functions (e.g., $u_0(x) = |x|^{2-n}-1$ for $n \ge 3$) whose gradients are not $L^p$ integrable at infinity. The critical space for such failure is $W^{1,p}$, with a one-dimensional kernel obstructing uniform estimates (Jiang et al., 2024).

Context	Critical Exponent $p$	Boundedness of $R$
Manifolds with ends	$p < \min_i n_i$	Bounded
	$p \ge \min_i n_i$	Unbounded
Exterior Lipschitz domains	$p < n$	Bounded
	$p \ge n$	Unbounded

5. Unboundedness for Singular, Fractal, and Irregular Measures

On $\mathbb{R}^d$ with singular measures of dimension $s$, the $s$-Riesz operator $R\mu = \mu * \frac{x}{|x|^{s+1}}$ is unbounded on $L^\infty$ for all non-integer $s \in (0, d)$. This is proven via Cantor-type decompositions, non-homogeneous Calderón–Zygmund theory, and maximal potential principles. For totally irregular measures, even adaptation to Schrödinger operators fails to restore boundedness on $L^2(\mu)$, including the presence of reverse-Hölder potentials in the operator (Eiderman et al., 2011, Bailey et al., 2020). The unboundedness results generalize to arbitrarily rough sets and non-doubling contexts.

6. The Dichotomy Principle and Construction of "Bad" Manifolds

The dichotomy in $L^p$ boundedness of the Riesz transform on Riemannian manifolds asserts: for each $d$ and $p$, either all $d$-manifolds possess a finite $L^p$ bound, or there exists a single $d$- or $d+1$-dimensional manifold where the operator is unbounded (Amenta et al., 2018). The construction involves gluing together sequences of manifolds with large Riesz norm via a backbone, transplanting regions of Riesz growth, and using heat kernel or Brownian motion comparison to ensure the operator norm blows up globally. This principle offers a practical route to verifying unboundedness in large generality.

7. $L^\infty$ and Endpoint Ill-Posedness

The Riesz operator is not bounded on $L^\infty(\mathbb{R}^n)$: explicit smooth compact-frequency-supported wave packets are constructed whose transforms grow logarithmically with frequency localization scale, demonstrating unbounded outputs from unit-norm inputs (Li et al., 22 Jan 2026). This endpoint failure is more severe than mere absence of strong boundedness; pressure-projection in fluid-type equations (e.g., Euler equations) becomes mildly ill-posed in $W^{1,\infty}$, with arbitrarily small initial data producing explosive derivatives in short time.

8. Reverse Riesz Operator and Counterexamples to Equivalence

On manifolds with ends, the reverse Riesz transform $R^* = \Delta^{-1/2} \nabla$ is bounded for all $1 < p < \infty$, in stark contrast with the original operator $R$ (He, 2024). This breaks the expected equivalence seen on doubling spaces with global Poincaré inequality. The geometric obstruction is the minimal-dimensional end dictating failure for $R$ beyond $p = n_*$, whereas $R^*$ is salvaged by harmonic annihilation and Hardy inequalities. On fractal graphs (e.g., graphical Sierpinski gasket), reverse Riesz is unbounded on $L^p$ for $1$p\ge2$ (Feneuil, 15 Feb 2025). This dichotomy further evidences the nuanced behavior emerging from non-Euclidean or non-doubling contexts.

9. Conclusion and Broader Implications

Unboundedness of the Riesz operator is a precise geometric and analytic marker for breakdown of regularity, dimension-matching, and analytic control in noncompact, singular, or fractal settings. The study of its threshold phenomena illuminates failure of Sobolev–Poincaré inequalities, the necessity of volume-doubling, the impact of irregular measures, and the singularities in spectral resolvent expansions. The sharpness of parametrices, Hardy–Hilbert inequalities, and explicit counterexamples together yield a comprehensive picture of where and how Riesz theory encounters its intrinsic limitations.