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Let ${\mu }_A$ be the singular measure on the Heisenberg group ${ℍ}^n$ supported on the graph of the quadratic function $\varphi \left(y\right)=y^{t}Ay$, where $A$ is a $2n\times 2n$ real symmetric matrix. If $det(2A\pm J)\ne 0$, we prove that the operator of convolution by ${\mu }_A$ on the right is bounded from $L^{\frac{(2n+2)}{(2n+1)}}\left({ℍ}^n\right)$ to $L^{2n+2}\left({ℍ}^n\right)$. We also study the type set of the measures $\mathrm{d}{\nu }_{\gamma }(y,s)=\eta \left(y\right){\left|y\right|}^{-\gamma }\mathrm{d}{\mu }_A(y,s)$, for $0\le \gamma <2n$, where $\eta$ is a cut-off function around the origin on ${ℝ}^{2n}$. Moreover, for $\gamma =0$ we characterize the type set of ${\nu }_0$.

Let $A_1,\cdots ,A_m$ be $n\times n$ real matrices such that for each $1\le i\le m,$ $A_i$ is invertible and $A_i-A_j$ is invertible for $i\ne j$. In this paper we study integral operators of the form $$Tf\left(x\right)=\int k_1(x-A_{1}y)k_2(x-A_{2}y)\cdots k_m(x-A_{m}y)f\left(y\right)\mathrm{d}y,$$ $k_i\left(y\right)={\sum }_{j\in \mathbb{Z}}{2}^{jn/q_i}{\varphi }_{i,j}\left(2^{j}y\right)$, $1\le q_i<\infty ,$ $1/q_1+1/q_2+\cdots +1/q_m=1-r,$ $0\le r<1,$ and ${\varphi }_{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:H^p\left({ℝ}^n\right)\to L^q\left({ℝ}^n\right)$ for $0<p<1/r$ and $1/q=1/p-r.$ We also show that we can not expect the $H^p$-$H^q$ boundedness of this kind of operators.