On isomorphism of some functional spaces under action of integro-differential operators

Abstract

In the paper we consider representations of the second kind for solutions to the linear general uniform first order elliptic system in the unit circle $D= \{z : |z| \leq 1\}$ written in terms of complex functions: \begin{equation*} \mathcal D w \equiv \partial_{\bar z} w + q_1(z) \partial_z w + q_2(z) \partial_{\bar z} \overline w +A(z)w+B(z) \overline w=R(z), \end{equation*} where $w=w(z)=u(z)+iv(z)$ is the sought complex function, $q_1(z)$ and $q_2(z)$ are given measurable complex functions satisfying the uniform ellipticity condition of the system: \begin{equation*} |q_1(z)| + |q_2(z)| \leq q_0 = {\rm const}<1,\, z\in \overline D, \end{equation*} and $A(z),\,B(z), \,R(z)\in L_p(\overline D)$, $p>2$, are also given complex functions. The representation of the second kind is based on the well–known Pompeiu's formula: if $w\in W^1_p(\overline D)$, $p>2$, then \begin{equation*} \displaystyle w(z) = \dfrac{1}{2 \pi i} \int\limits_{\Gamma} \dfrac{w(\zeta)}{\zeta-z}d \zeta - \dfrac{1}{\pi}\iint\limits_D \dfrac{\partial w}{\partial \bar z} \cdot \dfrac{d \xi d \eta}{\zeta-z}, \end{equation*} where $w(z) \in W^1_p(\overline D)$, $p>2$. Then for the solution $w(z)$ we can write the representation \begin{equation*} \Omega(w) = \dfrac{1}{2 \pi i} \int\limits_{\Gamma} \dfrac{w(\zeta)}{\zeta-z}d \zeta +TR(z) \end{equation*} where \begin{equation*} \Omega(w) \equiv w(z) + T ( q_1(z) \partial_z w + q_2(z) \partial_{\bar z} \overline w +A(z)w + B(z) \overline w). \end{equation*} Under appropriate assumptions about on coefficients we prove that $\Omega$ is the isomorphism of the spaces $C^k_\alpha (\overline D) $ and $W^k_p (\overline D) $, $k\geq $1, $0 <\alpha <$1, $p> $2. These results develop and complete B.V. Boyarsky's works, where representations of the first kind were obtained. Also this work complete author's results on representations of the second kind with more difficult operators. As an implication of the properties of the operator $\Omega$, we obtain apriori estimates for the norms $\|w\|_{C^{k+1}_{\alpha}(\overline D)}$ and $\|w\|_{W^{k}_{p}(\overline D)}$.