• Title, Summary, Keyword: Hardy space

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HARDY SPACE OF LOMMEL FUNCTIONS

  • Yagmur, Nihat
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.1035-1046
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    • 2015
  • In this work we present some geometric properties (like star-likeness and convexity of order ${\alpha}$ and also close-to-convexity of order ($1+{\alpha}$)/2) for normalized of Lommel functions of the first kind. In order to prove our main results, we use the technique of differential subordinations and some inequalities. Furthermore, we present some applications of convexity involving Lommel functions associated with the Hardy space of analytic functions, i.e., we obtain conditions for the function $h_{{\mu},{\upsilon}}(z)$ to belong to the Hardy space $H^p$.

MATRICES OF TOEPLITZ OPERATORS ON HARDY SPACES OVER BOUNDED DOMAINS

  • Chung, Young-Bok
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1421-1441
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    • 2017
  • We compute explicitly the matrix represented by the Toeplitz operator on the Hardy space over a smoothly finitely connected bounded domain in the plane with respect to special orthonormal bases consisting of the classical kernel functions for the space of square integrable functions and for the Hardy space. The Fourier coefficients of the symbol of the Toeplitz operator are obtained from zeroth row vectors and zeroth column vectors of the matrix. And we also find some condition for the product of two Toeplitz operators to be a Toeplitz operator in terms of matrices.

COMPUTATION OF THE MATRIX OF THE TOEPLITZ OPERATOR ON THE HARDY SPACE

  • Chung, Young-Bok
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1135-1143
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    • 2019
  • The matrix representation of the Toeplitz operator on the Hardy space with respect to a generalized orthonormal basis for the space of square integrable functions associated to a bounded simply connected region in the complex plane is completely computed in terms of only the Szegő kernel and the Garabedian kernels.

THE HARDY TYPE INEQUALITY ON METRIC MEASURE SPACES

  • Du, Feng;Mao, Jing;Wang, Qiaoling;Wu, Chuanxi
    • Journal of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1359-1380
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    • 2018
  • In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ($n{\geq}3$), then it has exactly the n-dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curvature on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n-dimensional Berwald space of non-negative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n.