Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(t\xi)=t\varrho(\xi)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $\delta>0$, we consider convolution operator {\Cal T}^{\delta} associated with the smooth cone type multipliers defined by \hat {{\Cal T}^{\delta} f}(\xi,\tau)= (1-\frac{\varrho(\xi)}{|\tau|} )^{\delta}_+\hat f (\xi,\tau), (\xi,\tau)\in {\Bbb R}^d \times \Bbb R. If the unit sphere $\Sigma_{\varrho}\fallingdotseq\{\xi\in {\Bbb R}^d : \varrho(\xi)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that {\Cal T}^{\delta(p)} maps from $H^p({\Bbb R}^{d+1})$, $0<p<1$, into weak-$L^p(\Gamma_{\gamma})$ for the critical index $\delta(p)=d(1/p -1/2)-1/2$, where $\Gamma_{\gamma}=\{(x,t)\in {\Bbb R}^d\times\Bbb R : |t|\geq\gamma |x|\}$ for $\gamma=\max\{\sup_{\varrho(\xi)\leq 1}|\xi|,1\}$. Moreover, we furnish a function $f\in\frak S({\Bbb R}^{d+1})$ such that \sup_{\lambda>0} \lambda^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminus\Gamma_{\gamma}} : |{\Cal T}_{\varrho}^{\delta(p)}f(x,t)|>\lambda\}|=\infty.Comment: 13 page

The OPAL Equation of State and Low Metallicity Isochrones

The Yale stellar evolution code has been modified to use the OPAL equation of state tables (Rogers 1994). Stellar models and isochrones were constructed for low metallicity systems ($-2.8 \le [Fe/H] \le -0.6$). Above M\sim 0.7\,\msun, the isochrones are very similar to those which are constructed using an equation of state which includes the analytical Debye-Huckel correction at high temperatures. The absolute magnitude of the main sequence turn-off (\mvto) with the OPAL or Debye-Huckel isochrones is about 0.06 magnitudes fainter, at a given age, than \mvto derived from isochrones which do not include the Debye-Huckel correction. As a consequence, globular clusters ages derived using \mvto are reduced by 6 -- 7\% as compared to the ages determined from the standard isochrones. Below M\sim 0.7\,\msun, the OPAL isochrones are systematically hotter (by approximately 0.04 in B-V) at a given magnitude as compared to the standard, or Debye-Huckel isochrones. However, the lower mass models fall out of the OPAL table range, and this could be the cause of the differences in the location of the lower main-sequences.Comment: to appear in ApJ, 8 pages LaTeX, uses aaspptwo.sty. Complete uuencoded postscript file (including figures) available from: ftp://ftp.cita.utoronto.ca/cita/chaboyer/papers/opal.u