Certain classes of harmonic functions with singularities on a quasiconformal arc. (English. Russian original) Zbl 0732.31003

Andrievskij, V. V.

Ukr. Math. J. 41, No. 4, 406-409 (1989); translation from Ukr. Mat. Zh. 41, No. 4, 465-468 (1989).

The present paper is devoted to the study of a problem concerned with the constructive description of classes of harmonic functions and is closely connected with studies initiated in the middle sixties by Dzyadyk. Let \(L\subset {\mathbb{C}}\) be a bounded quasiconformal arc, and let \(C_{\Delta}(L)\) be the class of functions which are real, continuous on \({\bar {\mathbb{C}}}\), and harmonic in \({\bar {\mathbb{C}}}\setminus L\), \(U(z,\delta)=\{\zeta: | \zeta -z| \leq \delta \}\), \(z\in {\mathbb{C}}\), \(\delta >0.\)
We denote by \(C^{\omega}_{k,\Delta}(L)\), where \(\omega\) (\(\delta\)), \(\delta >0\) is a nondecreasing function, \(\omega (+0)=0\), the class of functions \(u\in C_{\Delta}(L)\) satisfying the condition \(\omega_{k,\Delta}(u,\delta,{\mathbb{C}})\leq c\omega (\delta)\), \(\forall \delta >0\) for the k-th harmonic modulus of smoothness. We assume also that \(\omega\) (\(\delta\)) satisfies the relation \(\omega (t\delta)\leq c_ 1t^ k\omega (\delta)\), \(\forall t>1\), \(\forall \delta >0.\)
In an earlier paper [ibid. 40, No.1, 3-7 (1988; Zbl 0699.31002)] a constructive characteristic was obtained for the classes \(C^{\omega}_{1,\Delta}(L)\). In the present paper we generalize this result to the case of arbitrary \(k>1\).

MSC:

31A15	Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31C05	Harmonic, subharmonic, superharmonic functions on other spaces
30E10	Approximation in the complex plane
30C62	Quasiconformal mappings in the complex plane

Keywords:

harmonic functions; quasiconformal arc

Citations:

Zbl 0699.31002

References:

[1]	V. K. Dzyadyk, ?On analytic and harmonic transformations of functions and on approximation of harmonic functions,? Ukr. Mat. Zh.,19, No. 5, 33-57 (1967).
[2]	V. K. Dzyadyk, Introduction to the Theory of the Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977). · Zbl 0481.41001
[3]	L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Toronto, Ontario (1966).
[4]	M. Z. Dveirin, ?A theorem of Hardy-Littlewood in domains with quasiconformal boundaries and its applications to harmonic functions,? Sib. Mat. Zh.,27, No. 3, 68-73 (1986). · Zbl 0615.34065 · doi:10.1007/BF00969344
[5]	P. M. Tamrazov, Smoothness and Polynomial Approximations [in Russian], Naukova Dumka, Kiev (1975). · Zbl 0351.41004
[6]	V. V. Andrievskii, ?A constructive description of classes of harmonic functions with singularities on a quasiconformal arc,? Ukr. Mat. Zh.,40, No. 1, 3-7 (1988). · Zbl 0699.31002 · doi:10.1007/BF01056436
[7]	V. V. Andrievskii, ?Direct theorems in the theory of approximation on quasiconformal arcs,? Izv. Akad. Nauk SSSR, Ser. Mat.,44, No. 2, 243-261 (1980).