2,319 research outputs found

### 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

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