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In this paper, we study the transition densities of pure-jump symmetric Markov processes in $ {{\mathbb R}}^d$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-L\'evy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at the infinity., Comment: 50 pages

We consider a subordinate Brownian motion $X$ with Gaussian components when the scaling order of purely discontinuous part is between $0$ and $2$ including $2$. In this paper we establish sharp two-sided bounds for transition density of $X$ in ${\mathbb R}^d$ and $C^{1,1}$-open sets. As a corollary, we obtain a sharp Green function estimates., Comment: 30 pages, 1 figure