scholarly journals Toeplitz operators on Bergman spaces of polygonal domains

2019 ◽  
Vol 62 (4) ◽  
pp. 1115-1136 ◽  
Author(s):  
Paula Mannersalo
Keyword(s):  
Connected Domain ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Bergman Projection ◽  
Polygonal Domains ◽  
Simply Connected Domain ◽  
Whitney Decomposition ◽  
Polygonal Boundary ◽  
Simply Connected

AbstractWe study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

scholarly journals MOSER–TRUDINGER INEQUALITY ON CONFORMAL DISCS

2010 ◽  
Vol 12 (06) ◽  
pp. 1055-1068 ◽  
Author(s):  
G. MANCINI ◽  
K. SANDEEP
Keyword(s):  
Connected Domain ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Poincaré Metric ◽  
Simply Connected Domain ◽  
Trudinger Inequality ◽  
Necessary And Sufficient ◽  
Simply Connected

We prove that a sharp Moser–Trudinger inequality holds true on a conformal disc if and only if the metric is bounded from above by the Poincaré metric. We also derive necessary and sufficient conditions for the validity of a sharp Moser–Trudinger inequality on a simply connected domain in ℝ2.

scholarly journals Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sumin Kim ◽  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractIn this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator $T_{\varphi }$ T φ on the Bergman space $A^{2}(\mathbb{D})$ A 2 ( D ) with non-harmonic symbols under certain assumptions.

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

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