Abstract
By using the way of weight coefficient and the theory of operators, we define a Hilbert-type operator with a class of homogeneous kernels and obtain its norm. As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernels of -degree is established, and some particular cases are considered.
Highlights
In 1908, Weyl published the well-known Hilbert’s inequality as the following
For A B α β 1 in 10, inequality 37, it reduces to the equivalent result of 3.1 in this paper
For k x, y being −1-degree homogeneous, inequalities 1.15 reduce to 1.3 - 1.4 in the symmetric kernel
Summary
Received 15 September 2008; Accepted 20 February 2009 Recommended by Patricia J. Y. Wong By using the way of weight coefficient and the theory of operators, we define a Hilbert-type operator with a class of homogeneous kernels and obtain its norm. An extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernels of −λ-degree is established, and some particular cases are considered.
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