For
$m,n\,\in \,\mathbb{N},\,1\,<\,m\le \,n$
, we write
$n={{n}_{1}}+\cdots +{{n}_{m}}$
where
$\{{{n}_{1}},\ldots \,,{{n}_{m}}\}\,\subset \,\mathbb{N}$
. Let
${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$
be
$n\,\times \,n$
singular real matrices such that
$$\underset{i=1\,}{\overset{m}{\mathop{\oplus }}}\,\underset{1\le j\ne i\le m}{\mathop \bigcap }\,\,{{N}_{j}}\,=\,{{\mathbb{R}}^{n}},$$
where
${{N}_{j}}\,=\,\{x\,:\,{{A}_{j}}x\,=\,0\},\,\text{dim(}{{N}_{j}}\text{)}\,=\,n\,-\,{{n}_{j}}$
, and
${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$
is invertible. In this paper we study integral operators of the form
$${{T}_{r}}f(x)\,=\,{{\int }_{{{\mathbb{R}}^{n}}}}|x\,-\,{{A}_{1y}}{{|}^{-{{n}_{1}}+{{\alpha }_{{{1}_{\ldots }}}}}}|x\,-\,{{A}_{m}}y{{|}^{-{{n}_{m}}+{{\alpha }_{m}}}}f(y)dy,$$
${{n}_{1}}\,+\,\cdots \,+\,{{n}_{m}}\,=\,n$
,
$\frac{{{\alpha }_{1}}}{{{n}_{1}}}\,=\,\cdots \,=\,\frac{{{\alpha }_{m}}}{{{n}_{m}}}\,=\,r$
,
$0\,<\,r\,<\,1$
, and the matrices
${{A}_{i}}\text{ }\!\!'\!\!\text{ s}$
are as above. We obtain the
${{H}^{p}}({{\mathbb{R}}^{n}})\,-\,{{L}^{q}}({{\mathbb{R}}^{n}})$
boundedness of
${{T}_{r}}$
for
$0\,<\,p\,<\,\frac{1}{r}$
and
$\frac{1}{q}\,=\,\frac{1}{p}\,-\,r$
.