Hilbert-type integral operators: norms and inequalities. (English) Zbl 1246.31001
Pardalos, Panos M. (ed.) et al., Nonlinear analysis. Stability, approximation, and inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday. New York, NY: Springer (ISBN 978-1-4614-3497-9/hbk; 978-1-4614-3498-6/ebook). Springer Optimization and Its Applications 68, 771-859 (2012).
Summary: The well-known Hilbert inequality and Hardy-Hilbert inequality may be rewritten in the form of inequalities relating the Hilbert operator and the Hardy-Hilbert operator with their norms. These two operators are some particular kinds of Hilbert-type operators, which have played an important role in mathematical analysis and applications. In this chapter, by applying methods of real analysis and operator theory, we define general Hilbert-type integral operators and study six particular kinds of these operators with different measurable kernels in several normed spaces. The norms, equivalent inequalities, some particular examples, and compositions of two operators are considered. In Section 42.1, we define weight functions with some parameters and give two equivalent inequalities for general measurable kernels. Also, the norm of Hilbert-type integral operators is estimated. In Section 42.2 and Section 42.3, four kinds of Hilbert-type integral operators with particular kernels in the first quarter and in the whole plane are obtained. In Section 42.4, we define two kinds of operators with kernels of several variables and obtain their norms. In Section 42.5, two kinds of compositions of Hilbert-type integral operators are considered. The lemmas and theorems provide an extensive account for this kind of operators.
For the entire collection see [Zbl 1242.00056].
For the entire collection see [Zbl 1242.00056].
MSC:
| 31A10 | Integral representations, integral operators, integral equations methods in two dimensions |
| 31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |
| 45P05 | Integral operators |
| 47G10 | Integral operators |
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