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 Mat. Sb. (N.S.), 1988, Volume 136(178), Number 1(5), Pages 24–40 (Mi msb1726)

The $A$-integral and boundary values of analytic functions

T. S. Salimov

Abstract: Let $G$ be a simply connected bounded domain on the complex plane $\mathbf C$, let $\gamma=\partial G$, and assume that $\gamma$ is a closed rectifiable Jordan curve. Denote by $m$ the Lebesgue linear measure on $\gamma$. For a function $F$ analytic on $G$ and for $\alpha>1$ let $F_\alpha^*(t)=\sup\{|F(z)|:z\in G, |z-t|<\alpha\rho(z,\gamma)\}$, $t\in\gamma$, where $\rho(z,\gamma)$ is the Euclidean distance from $z$ to $\gamma$. It is proved that if for some $\alpha>2$
$$m\{t\in\gamma:F^*_\alpha(t)>\lambda\}=o(\lambda^{-1}),\qquad\lambda\to+\infty,$$
then $F$ has a finite nontangential boundary value $F(t)$ for almost all $t\in\gamma$, and
$$(A)\int_\gamma F(t) dt=0,$$
where the integral on the left-hand side is understood as an $A$-integral. It is also proved that under condition (1) the function $F$ is representable in $G$ by the Cauchy $A$-integral of its nontangential boundary values on $\gamma$. Further, if $\gamma$ is regular (i.e., $m\{t\in\gamma:|t-z|\leqslant r\}\leqslant Cr$ for all $z\in\mathbf C$ and $r>0$, where the constant $C$ is independent of $z$ and $r$), then these assertions are valid if condition (1) holds for some $\alpha>1$.
The question of representability of integrals of Cauchy type by Cauchy $A$-integrals is studied. In particular, well-known results of Ul'yanov on this question are carried over to the case of domains with a regular boundary. It is proved that the condition of regularity of the boundary cannot be weakened here.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 64:1, 23–39

Bibliographic databases:

UDC: 517.5
MSC: Primary 30E20; Secondary 30E25

Citation: T. S. Salimov, “The $A$-integral and boundary values of analytic functions”, Mat. Sb. (N.S.), 136(178):1(5) (1988), 24–40; Math. USSR-Sb., 64:1 (1989), 23–39

Citation in format AMSBIB
\Bibitem{Sal88}
\by T.~S.~Salimov
\paper The $A$-integral and boundary values of analytic functions
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 136(178)
\issue 1(5)
\pages 24--40
\mathnet{http://mi.mathnet.ru/msb1726}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=945898}
\zmath{https://zbmath.org/?q=an:0669.30031|0654.30031}
\transl
\jour Math. USSR-Sb.
\yr 1989
\vol 64
\issue 1
\pages 23--39
\crossref{https://doi.org/10.1070/SM1989v064n01ABEH003292}

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3. R. A. Aliyev, “Existence of angular boundary values and Cauchy–Green formula”, Zhurn. matem. fiz., anal., geom., 7:1 (2011), 3–18
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7. Aliev R.A., “on Laurent Coefficients of Cauchy Type Integrals of Finite Complex Measures”, Proc. Inst. Math. Mech., 42:2 (2016), 292–303
8. Aliev R.A., “Representability of Cauchy-type integrals of finite complex measures on the real axis in terms of their boundary values”, Complex Var. Elliptic Equ., 62:4 (2017), 536–553
9. Aliev R.A. Nebiyeva Kh. I., “The a-Integral and Restricted Ahlfors-Beurling Transform”, Integral Transform. Spec. Funct., 29:10 (2018), 820–830
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