Sharp boundedness criteria for nonnegative kernels on regular curves in the Heisenberg group

Abstract

In this paper we study singular integrals associated with the nonnegative kernels \[ K_\alpha(x,y,z)=\frac{|z|^{\alpha/2}}{\|(x,y,z)\|^{\alpha+1}}, \qquad \alpha>0, \] in the first Heisenberg group $\mathbb H$, where $\|\cdot\|$ denotes the Kor\'anyi norm. This family was introduced by Chousionis and Li. We first show that there exists a compact $1$-Ahlfors regular horizontal $C^1$ curve $E\subset\mathbb H$ such that the singular integrals associated with $K_\alpha$ are unbounded on $L^2(E)$ for every $0<\alpha<4$. This extends the counterexamples of Chousionis, Li and Zhang from $0<\alpha<2$ to the full range $0<\alpha<4$. Combined with the boundedness theorem of F\"assler and Orponen for $\alpha\ge4$, our result determines the exact range of exponents for boundedness on every $1$-Ahlfors regular curve. We also consider the more general kernels \[ K_\Phi(p)=\frac{1}{\|p\|} \Phi\!\left(\frac{|z|^{1/2}}{\|p\|}\right), \qquad p=(x,y,z)\neq0, \] where $\Phi:[0,1]\to[0,\infty)$ is continuous and nondecreasing, with $\Phi(0)=0$. We show that the associated singular integral is $L^2$-bounded on every $1$-Ahlfors regular curve if and only if $\Phi(u)=O(u^4)$ as $u\to0$. When this condition fails, we construct a compact $1$-Ahlfors regular horizontal $C^1$ curve on which the operator is unbounded. The curves in both constructions can be chosen as horizontal lifts of planar $C^1$ graphs with arbitrarily small slope.