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N). Introduce the function n ∗ ∗ − φm w(m) , vm φm z(m) ψ(z) := − m=1 analytic in the closure of D. Then the functions ψ, φk satisfy the R-linear boundary conditions ∗ , vk ψ(t) = φk (t) − φk (t) + φk w(k) |t − ak | = rk , k = 1, . . 4) because n ∗ ∗ − φm w(m) vm φm t(m) vk ψ(t) = −vk m=1 n ∗ ∗ − φm w(m) vm φm t(m) = −vk ∗ − vk vk φk (t) + vk vk φk w(k) m=1,m=k ∗ , = φk (t) − φk (t) + φk w(k) where tk∗ = rk2 rk2 + ak = + ak = rk eiθ + ak = t. t − ak rk eiθ + ak − ak ∗ ) = c + id then Note that if φk (w(k) k k ∗ vk ψ(t) = Re φk (t) + Im φk (t) − Re φk (t) + Im φk (t) + φk w(k) ∗ = 2 Im φk (t) + φk w(k) = 2 Im φk (t) + ck + idk .

N. N. V. V. N. Ospanov The purpose of this work is to answer this question. , there is a constant c > 0 such that f L1 (Q) ≤c f F (Q) , ∀f ∈ F (Q); (b) the set C ∞ (Q) of infinitely differentiable functions in Q is dense in F (Q); (c) if g ∈ F (Q) and ψ ∈ C ∞ (Q), then ψ · g ∈ F (Q); (d) if g ∈ F (Q), then |g| ∈ F (Q) and |g| F (Q) ≤ c1 g F (Q) . For example, Lp (Q) and the Lorentz space Lp,r (Q) (1 < p, r < ∞) are spaces of type F , where the norm of Lp,r (Q) is given in the following form (see [2]): u Lp,r (Q) +∞ = μ x ∈ Q : f (x) ≥ t 0 1/p t r dt t 1/r .

V. Mityushev, Poincaré α-series for classical Schottky groups, in Analytic Number Theory, Approximation Theory, and Special Functions (2014), pp. 827–852 Green Function of the Dirichlet Problem for the Laplacian and Inhomogeneous Boundary Value Problems for the Poisson Equation in a Punctured Domain Baltabek Kanguzhin and Niyaz Tokmagambetov Abstract The aim of this work is to present a new definition of the Green function of the Dirichlet problem for the Laplace equation prompted by the theory of ordinary differential equations and investigate correctly solvable boundary value problems for the Poisson equation in a punctured domain.

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