Hilbert-Type Integral Operators: Norms and Inequalities

The well known Hilbert inequality and Hardy–Hilbert inequality may be rewritten in the forms of inequalities relating Hilbert operator and 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 the methods of Real Analysis and Operator Theory, we define a general Hilbert-type integral operator and study six particular kinds of this operator with different measurable kernels in several normed spaces. The norms, equivalent inequalities, some particular examples, and compositions of two operators are considered. In Sect. 42.1, we define the weight functions with some parameters and give two equivalent inequalities with the general measurable kernels. Meanwhile, the norm of a Hilbert-type integral operator is estimated. In Sect. 42.2 and Sect. 42.3, four kinds of Hilbert-type integral operators with the particular kernels in the first quarter and in the whole plane are obtained. In Sect. 42.4, we define two kinds of operators with the kernels of multi-variables and obtain their norms. In Sect. 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.