Endpoint theory of rough homogeneous singular integrals beyond the L log L threshold: weak (1,1) bounds for odd and Hardy-space kernels, and L^2 boundedness on homogeneous groups

For a singular integral operator $T_\Omega$ on $\mathbb{R}^n$ whose kernel $\Omega((x-y)/|x-y|)/|x-y|^n$ is rough (no smoothness in the angular variable), the endpoint weak type $(1,1)$ bound is known when the angular kernel $\Omega$ lies in the Zygmund class $L\log L(\mathbb{S}^{n-1})$ and has vanishing integral (Seeger, 1996), and Tao (1999) proved the analogous weak $(1,1)$ result on arbitrary homogeneous groups, conditional on $L^2$ boundedness, which he also established for $L\log L$ kernels. The neighboring questions — whether weak $(1,1)$ holds for odd kernels merely in $L^1(\mathbb{S}^{n-1})$, or for even kernels in the atomic Hardy space $H^1(\mathbb{S}^{n-1})$, a class that strictly contains the mean-zero $L\log L(\mathbb{S}^{n-1})$ kernels — were explicitly posed as open by Tao in 1999 and remain open; likewise open is the $L^2$ boundedness of such operators on general homogeneous groups for mean-zero $H^1$ angular kernels, the residue of the source's conjecture that analogous $L^2$ results hold for arbitrary homogeneous groups after the odd-$L^1$ case (settled there by the method of rotations and the work of Ricci and Stein, as recorded in the source) and the $L\log L$ case (Theorem 1.1 of the source).

Background

A basic object of Calderón–Zygmund theory — the classical theory of singular integral operators — is the rough homogeneous singular integral. Let $n \ge 2$ and let $\Omega$ be an integrable function on the unit sphere $\mathbb{S}^{n-1}$, extended homogeneously of degree zero to $\mathbb{R}^n \setminus \{0\}$ by $\Omega(x) = \Omega(x/|x|)$. The associated operator is

$$T_\Omega f(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} \frac{\Omega((x-y)/|x-y|)}{|x-y|^n}\, f(y)\, dy ,$$

where $\mathrm{p.v.}$ ("principal value") denotes the limit, as $\epsilon \to 0$, of the integral over $|x-y| > \epsilon$; together with its maximal truncation $T_\Omega^* f(x) = \sup_{\epsilon > 0} \left| \int_{|x-y| > \epsilon} \frac{\Omega((x-y)/|x-y|)}{|x-y|^n} f(y)\, dy \right|$. "Rough" means that no smoothness is imposed on the angular part $\Omega$; only its size and cancellation are controlled. The standard cancellation hypothesis is $\int_{\mathbb{S}^{n-1}} \Omega(\theta)\, d\sigma(\theta) = 0$, where $d\sigma$ is surface measure on the sphere; it is automatic for odd kernels ($\Omega(-\theta) = -\Omega(\theta)$). The operator is said to be of weak type $(1,1)$ if $\lambda\, |\{ x : |T_\Omega f(x)| > \lambda \}| \le C_\Omega \| f \|_{L^1(\mathbb{R}^n)}$ for all $f \in L^1$ and $\lambda > 0$; by interpolation with $L^2$ (when available) this yields $L^p$ boundedness for $1 < p \le 2$, and it is the endpoint of the $L^p$ scale that fails to hold by general theory.

The classical history: Calderón and Zygmund (1956) proved $L^p$ boundedness, $1 < p < \infty$, when $\Omega$ is odd and in $L^1(\mathbb{S}^{n-1})$ — by the method of rotations, which averages the operator over all rotations of coordinates and thereby reduces odd kernels to one-dimensional Hilbert transforms along lines — or when $\Omega$ is even, in $L\log L(\mathbb{S}^{n-1})$ and of vanishing integral; here $L\log L$ is the Zygmund class with quasi-norm $\int_{\mathbb{S}^{n-1}} |\Omega| \log(2 + |\Omega|)\, d\sigma$. The $L^p$ theory was later extended to kernels in the atomic Hardy space $H^1(\mathbb{S}^{n-1})$ by Ricci and Weiss and by Connett (for the attribution see the introduction of the 2026 criterion paper of Liu, Liu and Shi cited below), where $H^1(\mathbb{S}^{n-1})$ is the Banach space of functions on the sphere admitting an atomic decomposition into "atoms" $a$ supported on spherical balls with $\| a \|_\infty \le 1/|B|$ and vanishing integral (for small supports). The class $H^1(\mathbb{S}^{n-1})$ strictly contains the mean-zero $L\log L$ class: every mean-zero $L\log L$ function on the sphere belongs to $H^1$, while atoms have uniformly bounded $H^1$ norm but unbounded $L\log L$ norm as their supports shrink. The weak $(1,1)$ problem proved far harder: in the plane it was settled around 1988 by Christ, by Hofmann, and by Christ and Rubio de Francia (covering $L^q$-type and $L\log L$ angular kernels); in all dimensions $n \ge 2$, Seeger (1996) proved that $T_\Omega$ is of weak type $(1,1)$ whenever $\Omega \in L\log L(\mathbb{S}^{n-1})$ has vanishing integral, by a microlocal decomposition of the kernel (decomposing it simultaneously by position and by frequency direction, that is, by the direction of the singularity). The $L\log L$ threshold has never been crossed: for odd $\Omega \in L^1(\mathbb{S}^{n-1})$, and for even $\Omega \in H^1(\mathbb{S}^{n-1})$, the weak $(1,1)$ question for $T_\Omega$ was stated as open in Tao's 1999 paper and remains open; the source also records that the corresponding $(H^1, L^1)$ conjecture is false by an example of Christ. The maximal truncation $T_\Omega^*$ is a later-literature variant which the source expressly defers ("We will not pursue these matters here"): the weak $(1,1)$ question for it was open even for $\Omega \in L^\infty(\mathbb{S}^{n-1})$ as recounted by Lerner (2019), and a 2025 preprint of Lai claims the weak $(1,1)$ bound for $T_\Omega^*$ for all mean-zero $L\log L$ kernels in $d \ge 2$, which would close that variant inside the $L\log L$ class while leaving the odd-$L^1$ and $H^1$ cases for $T_\Omega$ untouched. Related partial results in the literature include weak-type Orlicz endpoints, block-space or Lorentz-Orlicz kernel classes, sparse domination bounds (bounds obtained by representing the operator as a sum over sparse families of cubes), and results on restricted classes of homogeneous groups.

The non-Euclidean strand concerns homogeneous groups. A homogeneous group is a connected, simply connected nilpotent Lie group $H$ (identified with $\mathbb{R}^n$ through exponential coordinates, so that the identity is the origin and the group inversion is the sign flip $x^{-1} = -x$) equipped with a one-parameter family of contractive dilations $\delta_t(x_1, \ldots, x_n) = (t^{\alpha_1} x_1, \ldots, t^{\alpha_n} x_n)$, $t > 0$, under which the group operations transform covariantly; its homogeneous dimension is $Q = \alpha_1 + \cdots + \alpha_n$ (strictly larger than the topological dimension in the genuinely non-isotropic case, e.g. the Heisenberg group, the simplest non-commutative homogeneous group, which underlies sub-Riemannian geometry and quantum mechanics). Fixing a homogeneous norm $\rho$ (a continuous, inversion-symmetric function on $H$, homogeneous of degree one, vanishing only at the identity), the "sphere" $\Sigma = \{ x \in H : \rho(x) = 1 \}$ plays the role of $\mathbb{S}^{n-1}$. A homogeneous convolution operator has the form $T f(x) = \mathrm{p.v.} \int_H K(x y^{-1}) f(y)\, dy$ with kernel $K$ homogeneous of degree $-Q$ ($K(\delta_t x) = t^{-Q} K(x)$); its angular datum is the restriction $K_0 = K|_\Sigma$. On such groups the Euclidean Fourier transform and microlocal decompositions adapted to round spheres are unavailable, and the method of rotations — which in any case cannot be applied directly to weak $(1,1)$ boundedness — covers odd kernels only at the $L^2$ level: the source records that if $K_0$ is odd and in $L^1$, one can use the method of rotations and the work of Ricci and Stein on convolution operators on singular sets in homogeneous groups to obtain $L^2$ boundedness. Tao (1999) proved two results on arbitrary homogeneous groups: if $K_0 \in L\log L(\Sigma)$ has mean zero, then $T$ is bounded on $L^2(H)$ (Theorem 1.1); and if $K_0 \in L\log L(\Sigma)$ and $T$ is bounded on $L^2(H)$, then $T$ is of weak type $(1,1)$ (Theorem 1.2) — the latter by an iterated $(TT^*)^M$ method in the spirit of Christ and Rubio de Francia, which studies $M$-fold products of $T$ with its adjoint $T^*$, whose kernels are repeated convolutions of $K$ and gain decay with each factor. Together they give $L^p$ and weak $(1,1)$ bounds for all mean-zero $L\log L$ angular kernels on every homogeneous group, and the source conjectures that analogous $L^2$ results hold for arbitrary homogeneous groups. Whether $L^2$ boundedness holds for mean-zero $H^1(\Sigma)$ angular kernels — the Euclidean $L\log L$ condition having been relaxed to $H^1$ and beyond classically — is not known, and later work on homogeneous groups (Sato 2013; Fan and Li 2021) has remained within $L\log L$-type or $L^q$-type classes, all of which sit inside $L\log L$.

Problem Statement

The problem is the endpoint theory of rough homogeneous singular integrals for angular kernel classes beyond the $L\log L$ threshold. Two settings.

Part A (Euclidean space). Let $n \ge 2$, let $\Omega \in L^1(\mathbb{S}^{n-1})$ be extended homogeneously of degree zero, and let $T_\Omega$ be as defined in the background.

(A1) Odd kernels. If $\Omega$ is odd (so $\int_{\mathbb{S}^{n-1}} \Omega\, d\sigma = 0$ automatically) and $\Omega \in L^1(\mathbb{S}^{n-1})$, is $T_\Omega$ of weak type $(1,1)$ on $\mathbb{R}^n$?

(A2) Even Hardy kernels. If $\Omega$ is even and $\Omega \in H^1(\mathbb{S}^{n-1})$ (the atomic Hardy space, whose elements have vanishing integral and which strictly contains the mean-zero $L\log L(\mathbb{S}^{n-1})$ class), is $T_\Omega$ of weak type $(1,1)$ on $\mathbb{R}^n$?

Part B (homogeneous groups). Let $H$ be a homogeneous group with dilations $\delta_t$ and homogeneous dimension $Q$, let $\rho$ be a homogeneous norm with sphere $\Sigma = \{ x : \rho(x) = 1 \}$, and let $T f(x) = \mathrm{p.v.} \int_H K(x y^{-1}) f(y)\, dy$ with $K$ homogeneous of degree $-Q$ and angular kernel $K_0 = K|_\Sigma \in L^1(\Sigma)$.

(B) $L^2$ boundedness from cancellation beyond $L\log L$. If $K_0 \in H^1(\Sigma)$ has mean zero, is $T$ bounded on $L^2(H)$? (Known affirmatively for mean-zero $K_0 \in L\log L(\Sigma)$ by Theorem 1.1 of the source; for odd $K_0 \in L^1(\Sigma)$ — odd meaning $K_0(x^{-1}) = -K_0(x)$ for the group inversion, which in exponential coordinates is the sign flip $x^{-1} = -x$, the natural analogue on $\Sigma$ of the Euclidean condition $\Omega(-\theta) = -\Omega(\theta)$ — the source records that $L^2$ boundedness follows from the method of rotations and the work of Ricci and Stein on convolution operators on singular sets in homogeneous groups; the Euclidean analogue of the $H^1$ hypothesis implies $L^2$ boundedness classically.)

The common objective across all parts is to determine whether the endpoint ($L^2$ and weak $(1,1)$) Calderón–Zygmund theory of rough kernels extends past the $L\log L$ size threshold to the cancellation-driven classes (odd $L^1$; atomic $H^1$) for which the $L^p$ theory, $1 < p < \infty$, is classical.

References

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