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Let $L:= -a(x) (-\Delta)^{\alpha/2}+ (b(x), \nabla)$, where $\alpha\in(0,2)$, and $a:\rd\to (0,\infty)$, $b: \rd\to \rd$. Under certain regularityassumptions on the coefficients $a$ and $b$, we associate with the$C_\infty(\rd)$-closure of $(L, C_\infty^2(\rd))$ a Feller Markov process $X$,which possesses a transition probability density $p_t(x,y)$.  To construct this transition probability density and to obtain the two-sidedestimates on it, we develop a new version of the parametrix method, whichallows us to handle the case $0<\alpha\leq 1$ and $b\neq 0$, i.e. when thegradient part of the generator is not dominated by the jump part..

Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise