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Ufimsk. Mat. Zh., 2019, Volume 11, Issue 1, Pages 39–60 (Mi ufa459)  

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

S. B. Klimentovab

a Southern Federal University, Milchakov str. 8-a, 344090, Rostov-on-Don, Russia
b Southern Mathematical Institute, Vatutin str. 53, 362027, Vladikavkaz, Russia

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 $D= ż: |z| \leq 1\}$ 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 = 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 $\|w\|_{W^{k}_{p}(\overline D)}$.

Keywords: general elliptic first order system, representation of the second kind.

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English version:
Ufa Mathematical Journal, 2019, 11:1, 42–62 (PDF, 495 kB); https://doi.org/10.13108/2019-11-1-42

Bibliographic databases:

UDC: 517.518.234 + 517.548.3
MSC: 35C15
Received: 02.06.2017

Citation: S. B. Klimentov, “On isomorphism of some functional spaces under action of integro-differential operators”, Ufimsk. Mat. Zh., 11:1 (2019), 39–60; Ufa Math. J., 11:1 (2019), 42–62

Citation in format AMSBIB
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\by S.~B.~Klimentov
\paper On isomorphism of some functional spaces under action of integro-differential operators
\jour Ufimsk. Mat. Zh.
\yr 2019
\vol 11
\issue 1
\pages 39--60
\mathnet{http://mi.mathnet.ru/ufa459}
\transl
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 1
\pages 42--62
\crossref{https://doi.org/10.13108/2019-11-1-42}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85066016670}


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