  We study smoothness of densities for the solutions of SDEs whose coefficients
are smooth and nondegenerate only on an open domain $D$. We prove that a smooth
density exists on $D$ and give upper bounds for this density. Under some
additional conditions (mainly dealing with the growth of the coefficients and
their derivatives), we formulate upper bounds that are suitable to obtain
asymptotic estimates of the density for large values of the state variable
("tail" estimates). These results specify and extend some results by Kusuoka
and Stroock [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1--76], but our
approach is substantially different and based on a technique to estimate the
Fourier transform inspired from Fournier [Electron. J. Probab. 13 (2008)
135--156] and Bally [Integration by parts formula for locally smooth laws and
applications to equations with jumps I (2007) The Royal Swedish Academy of
Sciences]. This study is motivated by existing models for financial securities
which rely on SDEs with non-Lipschitz coefficients. Indeed, we apply our
results to a square root-type diffusion (CIR or CEV) with coefficients
depending on the state variable, that is, a situation where standard techniques
for density estimation based on Malliavin calculus do not apply. We establish
the existence of a smooth density, for which we give exponential estimates and
study the behavior at the origin (the singular point).
