Abstract. Let $\rho\in C^\infty({\msbm R}^d\setminus\{0\} )$ be a distance function, homogeneous with respect to a dilation group $(\exp t\log P)_{t>0}$. For $f\in L^1({\msbm R}^d)$ we consider the pointwise behavior of generalized Riesz means for the Fourier integral, defined by $\widehat{S^t_{\rho,\lambda } f}=(1-\rho(\xi )/t)^\lambda_+\widehat f$. Let ${\eufm M}_{\rho,\lambda }$ be the associated maximal operator. If $P$ is a multiple of the identity then it is known that $\eufm M_{\rho,\lambda }$ is of weak type $(1,1)$ if $\lambda >(d-1)/2$. We show that if $P$ is not a multiple of the identity then for suitable $\rho $ the weak type inequality may fail to hold for $\lambda < d/2$; moreover $L^p$ boundedness of $\eufm M_{\rho,\lambda }$ may fail to hold if $\lambda\le d(1/p-1/2)$. Sharp results are discussed for the case $d=2$, under an additional finite type assumption.

AMS Subject Classification (1991): 42B25, 42B15

Keyword(s): Quasiradial multipliers, Riesz means, pointwise convergence, maximal operators

Received December 11, 1995. (Registered under 5718/2009.)