Let
${{\left\{ {{A}_{t}} \right\}}_{t>0}}$
be the dilation group in
${{\mathbb{R}}^{n}}$
generated by the infinitesimal generator
$M$
where
${{A}_{t}}\,=\,\exp \left( M\,\log \,t \right)$
, and let
$\varrho \,\in \,{{C}^{\infty }}\left( \mathbb{R}{{}^{n\,}}\backslash \,\left\{ 0 \right\} \right)$
be a
${{A}_{t}}$
-homogeneous distance function defined on
${{\mathbb{R}}^{n}}$
. For
$f\,\in \,\mathfrak{S}\left( \mathbb{R}{{}^{n}} \right)$
, we define the maximal quasiradial Bochner-Riesz operator
$\mathfrak{M}_{\varrho }^{\delta }$
of index
$\delta \,>\,0$
by
$$\mathfrak{M}_{\varrho }^{\delta }f\left( x \right)\,=\,\underset{t>0}{\mathop{\sup }}\,\left| {{\mathcal{F}}^{-1}}\left[ \left( 1-{\varrho }/{t}\; \right)_{+}^{\delta }\hat{f} \right]\left( x \right) \right|.$$
If
${{A}_{t\,}}=\,tI$
and
$\left\{ \xi \,\in \,{{\mathbb{R}}^{n\,}}\,|\,\varrho \left( \xi \right)\,=\,1 \right\}$
is a smooth convex hypersurface of finite type, then we prove in an extremely easy way that
$\mathfrak{M}_{\varrho }^{\delta }$
is well defined on
${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$
when
$\delta \,=\,n(1/p\,-\,1/2)\,-\,1/2$
and
$0\,<\,p\,<\,1$
; moreover, it is a bounded operator from
${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$
into
${{L}^{p,\infty }}\left( {{\mathbb{R}}^{n}} \right)$
.
If
${{A}_{t}}\,=\,tI$
and
$\varrho \,\in \,{{C}^{\infty }}\left( \mathbb{R}{{}^{n\,}}\backslash \,\left\{ 0 \right\} \right)$
, we also prove that
$\mathfrak{M}_{\varrho }^{\delta }$
is a bounded operator from
${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$
into
${{L}^{p}}\,\left( {{\mathbb{R}}^{n}} \right)$
when
$\delta \,>\,n(1/p\,-\,1/2)\,-\,1/2$
and
$0\,<\,p\,<\,1$
.