Let $\Gamma$ be a rectifiable Jordan curve in the finite complex plane $\mathbb C$ which is regular in the sense of Ahlfors and David. Denote by $L^2_C (\Gamma)$ the space of all complex-valued functions on $\Gamma$ which are square integrable w.r. to the arc-length on $\Gamma$. Let $L^2(\Gamma)$ stand for the space of all real-valued functions in $L^2_C (\Gamma)$ and put
\[ L^2_0 (\Gamma) = \lbrace h \in L^2 (\Gamma)\; \int _{\Gamma} h(\zeta ) |\mathrm{d}\zeta | =0\rbrace. \] Since the Cauchy singular operator is bounded on $L^2_C (\Gamma)$, the Neumann-Poincaré operator $C_1^{\Gamma}$ sending each $h \in L^2 (\Gamma)$ into \[ C_1^{\Gamma} h(\zeta _0) := \Re (\pi \mathrm{i})^{-1} \mathop {\mathrm P. V.}\int _{\Gamma} \frac{h(\zeta )}{\zeta -\zeta _0} \mathrm{d}\zeta , \quad \zeta _0 \in \Gamma , \] is bounded on $L^2(\Gamma)$. We show that the inclusion
\[ C_1^{\Gamma} (L^2_0 (\Gamma)) \subset L^2_0 (\Gamma)
\] characterizes the circle in the class of all $AD$-regular Jordan curves $\Gamma$.
We prove and discuss some new (Hp,Lp)-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q_{k}:k\geqslant 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund means tn with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence {q_{k}:k\geqslant 0} in an appropriate way., István Blahota, Lars-Erik Persson, Giorgi Tephnadze., and Obsahuje seznam literatury
We study $n$-dimensional $QR$-submanifolds of $QR$-dimension $(p-1)$ immersed in a quaternionic space form $QP^{(n+p)/4}(c)$, $c\geqq 0$, and, in particular, determine such submanifolds with the induced normal almost contact $3$-structure.
New results are proved on the maximum number of isolated $T$-periodic solutions (limit cycles) of a first order polynomial differential equation with periodic coefficients. The exponents of the polynomial may be negative. The results are compared with the available literature and applied to a class of polynomial systems on the cylinder.
The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense dx(t) = dA0(t) · x(t) + df0(t), x(t0) = c0 (t ∈ I) with a unique solution x0 is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems dx(t) = dAk(t) · x(t) + dfk(t), x(tk) = ck (k = 1, 2, . . .) to have a unique solution xk for any sufficiently large k such that xk(t) → x0(t) uniformly on I. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.