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

This article is cited in 9 scientific papers (total in 9 papers)

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 $\alpha>2$
\begin{equation} m\{t\in\gamma:F^*_\alpha(t)>\lambda\}=o(\lambda^{-1}),\qquad\lambda\to+\infty, \end{equation}
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
Received: 29.06.1987

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
\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
\jour Math. USSR-Sb.
\yr 1989
\vol 64
\issue 1
\pages 23--39

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    This publication is cited in the following articles:
    1. A. V. Rybkin, “The spectral shift function, the characteristic function of a contraction, and a generalized integral”, Russian Acad. Sci. Sb. Math., 83:1 (1995), 237–281  mathnet  crossref  mathscinet  zmath  isi
    2. Harlouchet I., “Cauchy Trace for Certain Locally Integrable Functions on a Bounded Open Set of C”, Publ. Mat., 48:1 (2004), 69–102  mathscinet  zmath  isi
    3. R. A. Aliyev, “Existence of angular boundary values and Cauchy–Green formula”, Zhurn. matem. fiz., anal., geom., 7:1 (2011), 3–18  mathnet  mathscinet  zmath  elib
    4. R. A. Aliyev, “$N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures”, Sb. Math., 205:7 (2014), 913–935  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. R.A.. Aliev, “On Properties of Hilbert Transform of Finite Complex Measures”, Complex Anal. Oper. Theory, 2015  crossref
    6. Aliev R.A., “Riesz'S Equality For the Hilbert Transform of the Finite Complex Measures”, Azerbaijan J. Math., 6:1 (2016), 126–135  mathscinet  zmath  isi
    7. Aliev R.A., “on Laurent Coefficients of Cauchy Type Integrals of Finite Complex Measures”, Proc. Inst. Math. Mech., 42:2 (2016), 292–303  isi
    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  crossref  mathscinet  zmath  isi  scopus
    9. Aliev R.A. Nebiyeva Kh. I., “The a-Integral and Restricted Ahlfors-Beurling Transform”, Integral Transform. Spec. Funct., 29:10 (2018), 820–830  crossref  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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