shall denote the convolution operator defined by $(G*\varphi)(x)=\int_{\mathbb{R}^n} G(x-y)\varphi(y){\mathrm{d}}{y}$.

[@Henry1981 Section 1.6], [@Stein1970 Chapter 5] \[A-sectorial\] The Laplace operator $-\Delta $ is sectorial in the Banach space $X=L^{p}(\mathbb{R}^{n})$ of Lebesgue $p$-integrable functions for any $p\ge 1$ and $n\ge 1$. Its domain $D(A)$ is embedded into the Sobolev space $W^{2,p}(\mathbb{R}^{n})$. The fractional power space $X^\gamma, \gamma>0,$ is the space of Bessel potentials: $X^\gamma={\mathscr L}^p_{2\gamma}(\mathbb{R}^n):=\{ G_{2\gamma}*\varphi, \ \varphi\in L^p({\mathbb{R}}^n)\}$ where $$G_{2\gamma}(x) = \frac{(4\pi)^{-n/2}}{\Gamma(\gamma)}\int_0^\infty \xi^{-1+(2\gamma-n)/2} e^{-(\xi +\Vert x\Vert^2/(4\xi))}{\mathrm{d}}\xi$$ is the Bessel potential function. The norm of $u=G_{2\gamma}*\varphi$ is given by $\Vert u\Vert_{X^\gamma}=\Vert \varphi\Vert_{L^p}$. The fractional power space $X^\gamma$ is continuously embedded into the fractional Sobolev-Slobodeckii space $W^{2\gamma,p}({\mathbb{R}}^n)$.

Lemma \[A-sectorial\] was proven in [@Henry1981 Section 1.6], [@Stein1970 Chapter 5]. The idea of the proof of sectoriality of the Laplace operator $-\Delta $ in the Banach space $X=L^p(\mathbb{R}^n)$ is based on estimation of the resolvent operator $(\lambda -\Delta)^{-1}$ in the $L^p$ norm. The rest of the proof of Lemma \[A-sectorial\] is based on the analysis of the Fourier transform of the equation $(\lambda-\Delta)u=f$. The Fourier transform $\hat u$ of its solution is given by $\hat u(\xi) = (\lambda +|\xi|^2)^{-1}\hat f(\xi)$. The function $G_\alpha$ is then constructed by means of the inverse Fourier transform of $\hat 