Abstract
Let X be a space of functions that are analytic in the unit disk and have boundary values satisfying some additional smoothness conditions (X may be the space of functions with derivative in the Hardy space H 1, or analytic Besov space B 1/pp,1 ), and let E be a non-Carleson subset, i.e., L = – ∫ log dist(t, E)dm < ∞. For f ∈ X, we estimate |f(z)| by a quantity depending on || f ||x under the assumption that f vanishes on E. Simultaneously we obtain a formula that reconstructs f (z) at an arbitrary point z starting with the restriction of f to a Carleson subset E. This is an analogue of the Carleman-Golusin-Krylov formula for the functions in the Hardy space H 1 and the sets E of positive Lebesgue measure on the circle.
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Videnskii, I. (2000). Carleman Formula for Some Spaces of Functions Analytic in the Disk and Smooth in its Closure. In: Havin, V.P., Nikolski, N.K. (eds) Complex Analysis, Operators, and Related Topics. Operator Theory: Advances and Applications, vol 113. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8378-8_32
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DOI: https://doi.org/10.1007/978-3-0348-8378-8_32
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