$0<\alpha<2,\quad
\alpha{\ne}1$ that* $$\nu^{\prime}(y){\leq}C|y|^{-\alpha-1},\quad$$ *Proof.* The function $\nu(y)$ has the form $$\nu(y)=\int_{-\infty}^{y}{\nu}^{\prime}(t)dt1_{y<0}-\int_{y}^{\infty}{\nu}^{\prime}(t)dt1_{y>0}.$$First we shall consider the case, when $1<\alpha<2$, and introduce the function $$k_{0}(y)=\int_{-\infty}^{y}(y-t){\nu}^{\prime}(t)dt1_{y<0}-\int_{y}^{\infty}(y-t){\nu}^{\prime}(t)dt1_{y>0}.$$We obtain the relation $$k(y)=k_{0}(y)+(\gamma-{\Gamma})\mathrm{p}_{0}(y),\quad
1<\alpha<2,$$ where $k_{0}(y)$ and $p_{0}(y)$ are defined by (2.27) and (2.30) respectively. The constant $\Gamma$ is defined by the relation: $$\Gamma=k_{0}(-1)-k_{0}^{\prime}(-1)-k_{0}(-1)+k_{0}^{\prime}(1),\quad 1<\alpha<2,$$It follows from (2.28)-(2.30) that the conditions of Lemma 2.2 are fulfilled. Hence the proposition is true when $1<\alpha<2.$ Let us consider the case when $0<\alpha<1$. As in the previous case the function $\nu(x)$ is defined by relation (2.29). We introduce the functions $$k_{0}(y)=\int_{-\infty}^{y}\nu^{\prime}(t)dt\,y+\int_{y}^{0}\nu^{\prime}(t)tdt,
\quad y<0,$$ $$k_{0}(y)=-\int_{y}^{\infty}\nu^{\prime}(t)dt\,y-\int_{0}^{y}\nu^{\prime}(t)tdt,
\quad y>0,$$ and $$k(y)=k_{0}(y)+\gamma\mathrm{p}_{0}(y)\quad
0<\alpha<1.$$In view of (2.28) and (2.33),(2.34) the conditions of Lemma 2.1 are fulfilled. Hence the proposition is proved.\
**Corollary 2.1.** *If condition $(2.28)$ is fulfilled then* $$k_{0}(y){\geq}0,\quad
-\infty<y<\infty, \quad 1<\alpha<2,$$ $$k_{0}(y){\leq}0,\quad -\infty<y<\infty, \quad
0<\alpha<1.$$ Let us consider the important case when $\alpha=1.$\
**Proposition 2.2.** *The generator $L$ of the Levy process $X_{t}$ admits the convolution type representation $(2.1),(2.2)$ if there exist such $C>0$ and $m>0$ that* $$\nu^{\prime}(y){\leq}C|y|^{-2}e^{-m|y|}.$$ *Proof.* Using formulas (2.29)-(2.32) we see that the conditions of Lemma 2.2 are fulfilled. The proposition is proved.\
**Example 2.1.** The stable processes.\
For the stable processes we have $A=0,\quad
\gamma=\overline{\gamma}$ and $$\nu^{\prime}(y)=|y|^{-\alpha-1}(C_{1}1_{y<0}+C_{2}1_{y>0}),$$where 