S_{a,\phi} \subset \mathbb{C}\setminus\sigma(A)$.

In what follows, we shall investigate the partial-integral differential equation (\[problem\_transformed\]) in the framework of the so-called Bessel potential spaces. These spaces represent natural extension of the classical Sobolev spaces $W^{k,p}(\mathbb{R})$ where the order $k$ may attain the discrete values only, i.e. the distributional derivatives up to the order $k$ belong to the Lebesgue space $L^p(\mathbb{R})$. Bessel potential spaces represent a continuous scale of fractional powers, and allow for a finer formulation of results in comparison to the classical Sobolev spaces $W^{k,p}(\mathbb{R}), k\in\mathbb{N}$.

It is well known that that if $A$ is a sectorial operator then $-A$ is an infinitesimal generator of an analytic semigroup $S(t)=\left\{e^{-A t}, t\geq 0\right\}$ (cf. [@Henry1981]). If $X$ is a Banach space then we can define a scale of fractional power spaces $\{X^{\gamma}\}_{\gamma\ge 0}$ in the following way: $$X^{\gamma}= D(A^{\gamma})= Range(A^{-\gamma})=\left\{u\in X:\  \exists \varphi\in X,  u=A^{-\gamma}\varphi\right\},$$ where, for any $\gamma >0$, the operator $A^{-\gamma}$ is defined by virtue of the Gamma function, i.e. $A^{-\gamma}=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty} \xi^{\gamma-1}e^{-A \xi} {\mathrm{d}}\xi$. The norm is defined as $\Vert u\Vert_{X^\gamma}=\Vert A^{\gamma}u\Vert_X=\Vert \varphi\Vert_X$. Note that $X^0=X$, $X^1=D(A)$, and $X^1\equiv D(A) \hookrightarrow X^{\gamma_1} \hookrightarrow X^{\gamma_2} \hookrightarrow X^0\equiv X$, for any $0\le \gamma_2\le \gamma_1\le 1$.

In what follows, by $G*\varphi$ we 