We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces \(A^2_{\alpha}\), \(−1 < \alpha < \infty\). These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.
We estimate the Gauss curvature of nonparametric minimal surfaces over the two-slit plane \(\mathbb{C}\setminus ((-\infty,-1]\cup [1,\infty))\) at points above the interval \((-1, 1)\).
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We consider typically real harmonic univalent functions in the unit disk 𝔻 whose range is the complex plane slit along infinite intervals on each of the lines x ± ib, b > 0. They are obtained via the shear construction of conformal mappings of 𝔻 onto the plane without two or four half-lines symmetric with respect to the real axis.
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We prove a sufficient condition for products of Toeplitz operators $T_fT_{ḡ}$, where f,g are square integrable holomorphic functions in the unit ball in ℂⁿ, to be bounded on the weighted Bergman space. This condition slightly improves the result obtained by K. Stroethoff and D. Zheng. The analogous condition for boundedness of products of Hankel operators $H_fH*_g$ is also given.
We give new characterizations of the analytic Besov spaces \(B_p\) on the unit ball \(\mathbb{B}\) of \(\mathbb{C}^n\) in terms of oscillations and integral means over some Euclidian balls contained in \(\mathbb{B}\).
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