  In this paper, we consider a large class of purely discontinuous rotationally
symmetric Levy processes. We establish sharp two-sided estimates for the
transition densities of such processes killed upon leaving an open set D. When
D is a \kappa-fat open set, the sharp two-sided estimates are given in terms of
surviving probabilities and the global transition density of the Levy process.
When D is a C^{1, 1} open set and the Levy exponent of the process is given by
\Psi(\xi)= \phi(|\xi|^2) with \phi being a complete Bernstein function
satisfying a mild growth condition at infinity, our two-sided estimates are
explicit in terms of \Psi, the distance function to the boundary of D and the
jumping kernel of X, which give an affirmative answer to the conjecture posted
in [Potential Anal., 36 (2012) 235-261]. Our results are the first sharp
two-sided Dirichlet heat kernel estimates for a large class of symmetric Levy
processes with general Levy exponents. We also derive an explicit lower bound
estimate for symmetric Levy processes on R^d in terms of their Levy exponents.
