Weak type estimates of the maximal quasiradial Bochner-Riesz operator on certain hardy spaces

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