  Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t)
evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for
some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $\lambda_0(g_0)$ be the
first eigenvalue of the operator $-\Delta_{g_0} +\frac{R(g_0)}{4}$ with respect
to g_0. We extend a recent result of R. Ye and prove uniform logarithmic
Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for
any $n\ge 2$ when either $T<\infty$ or $\lambda_0(g_0)>0$. As a consequence we
extend Perelman's local $\kappa$-noncollapsing result along the Ricci flow for
any $n\ge 2$ in terms of upper bound for the scalar curvature when either
$T<\infty$ or $\lambda_0(g_0)>0$.
