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

In this paper we study singular integrals associated with the nonnegative kernels $$K_{\alpha }(x,y,z)=\frac{\mathrm{\mid }z{\mathrm{\mid }}^{\alpha \mathrm{/}2}}{\mathrm{\parallel }(x,y,z){\mathrm{\parallel }}^{\alpha +1}},\alpha >0,$$ in the first Heisenberg group $\mathbb{H}$, where $\mathrm{\parallel }\cdot \mathrm{\parallel }$ denotes the Korányi 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 Fässler and Orponen for $\alpha \ge 4$, our result determines the exact range of exponents for boundedness on every $1$-Ahlfors regular curve. We also consider the more general kernels $$K_{\mathrm{\Phi }}(p)=\frac{1}{\mathrm{\parallel }p\mathrm{\parallel }}\mathrm{\Phi }\text{\hspace{0.17em}⁣}\left(\frac{\mathrm{\mid }z{\mathrm{\mid }}^{1\mathrm{/}2}}{\mathrm{\parallel }p\mathrm{\parallel }}\right),p=(x,y,z)\mathrm{\ne }0,$$ where $\mathrm{\Phi }:[0,1]\to [0,\mathrm{\infty })$ is continuous and nondecreasing, with $\mathrm{\Phi }(0)=0$. We show that the associated singular integral is $L^2$-bounded on every $1$-Ahlfors regular curve if and only if $\mathrm{\Phi }(u)=O(u^4)$ as $u\to 0$. 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.