, (2.16) and the equality $J=J_{1}+J_{2}.$ The lemma is proved.\
**Lemma 2.2.** *Let the following conditions be fulfilled.\
1. The function $\nu(x)$ satisfies conditions $(2.4)$ and $(2.5)$ of Lemma $2.1$.\
2. For arbitrary $M\quad(0<M<\infty)$ we have $$\int_{-M}^{M}|k(x)|dx<\infty,\quad
\int_{-M}^{M}|x{\nu}(x)|dx<\infty,$$where* $$k^{\prime}(x)=\nu(x),\quad x{\ne}0.$$ 3. $$xk(x){\to}0,\quad x{\to}0;\quad x^{2}\nu(x){\to}0,\quad
x{\to}0.$$ *Then the equality* $$J=\int_{-\infty}^{\infty}[f(y+x)-f(x)-y\frac{df(x)}{dx}1_{D(y)}]{\nu}^{\prime}(y)dy
+{\Gamma}f^{\prime}(x),$$ *is true, where $\Gamma=\overline{\Gamma}$ and* $f(x){\in}C_{c}$.\
*Proof.* From (2.11) we obtain the relation $$J_{1}=f^{\prime}(x)\gamma_{1}-\int_{x-1}^{x}[f^{\prime}(y)-f^{\prime}(x)]k^{\prime}(y-x)dy-
\int_{-M}^{x-1}f^{\prime}(y)k^{\prime}(y-x)dy,$$ where $\gamma_{1}=k(-1).$ We introduce the notations $$P_{1}(x,y)=f(y)-f(x)-(y-x)f^{\prime}(x),\quad
P_{2}(x,y)=f(y)-f(x).$$Integrating by parts (2.21) and passing to the limit when $M{\to}\infty$ we deduce that $$J_{1}=f^{\prime}(x)\gamma_{2}+\int_{x-1}^{x}P_{1}(x,y)k^{\prime\prime}(y-x)dy
+\int_{-M}^{x-1}P_{2}(x,y)k^{\prime\prime}(y-x)dy,$$ where $\gamma_{2}=k(-1)-k^{\prime}(-1).$ It follows from (2.22) and (2.23) that $$J_{1}=\int_{-\infty}^{x}[f(y+x)-f(x)-y\frac{df(x)}{dx}1_{D(y)}]{\nu}^{\prime}(y)dy
+{\gamma}_{2}f^{\prime}(x).$$In the same way it can be proved that $$J_{2}=\int_{x}^{\infty}[f(y+x)-f(x)-y\frac{df(x)}{dx}1_{D(y)}]{\nu}^{\prime}(y)dy
+{\gamma}_{3}f^{\prime}(x),$$where $\gamma_{3}=-k(1)+k^{\prime}(1).$ The relation (2.20) follows directly from (2.24) and (2.25). Here $\Gamma=\gamma_{2}+\gamma_{3}.$ The lemma is proved.\
**Remark 2.1.** The operator $L_{0}f=\frac{d}{dx}f$ can be represented in form (2.1), (2.2), where $$S_{0}f=\int_{-\infty}^{\infty}p_{0}(y-x)f(y)dy,$$ $$p_{0}(x)=\frac{1}{2}\,\mathrm{sign}(x).$$\
From Lemmas 2.1 , 2.2 and Remark 2.1 we deduce the following assertion.\
**Theorem 2.1.** *Let the conditions of Lemma $2.1$ or Lemma $2.2$ be fulfilled . Then the corresponding operator $L$ has a convolution type form* (2.1),(2.2).\
**Proposition 2.1.** *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 