of wide class of processes for which such property holds true. Furthermore, if we restrict ourselves to the special case of spectrally negative Lévy processes then due to its specific structure, we are able to prove sharp two-sided estimates on both sides of the interval and for any $t>0$ (see Corollary \[cor:est\_spec\_neg\]). To our best knowledge, such results have not been obtained before. Apart from its own value, they can be used, together with heat kernel estimates ([@GS2019]) for instance for estimation of the Hausdorff dimension of the inverse images of Lévy processes (see [@Park2019]). We also remark that although our main object to operate with is the real part of the characteristic exponent, one can work with the tail of the Lévy measure instead, since in view of [@GLT18 Proposition 3.8], scaling property of the latter implies scaling of the former.

The first studies on the first hitting time of a point or a compact set concerned $\alpha$-stable processes. The asymptotic behaviour in the case of recurrent $\alpha$-stable process, i.e. $1<\alpha {\leqslant}2$, for arbitrary compact sets, was derived in [@Port67]. Next, in [@YYY79] the authors discuss the law of the first hitting time of a point for the symmetric 