On a Hilbert-Type Operator with a Class of Homogeneous Kernels
© Bicheng Yang 2009
Received: 15 September 2008
Accepted: 20 February 2009
Published: 4 March 2009
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.
where the constant factor is the best possible. We named (1.2) Hardy-Hilbert's inequality. In 1934, Hardy et al.  gave some applications of (1.1)-(1.2) and a basic theorem with the general kernel (see [3, Theorem 318]).
where the constant factors and are the best possible.
Hardy did not prove this theorem in . In particular, we find some classical Hilbert-type inequalities as,
(i)for in (1.3), it reduces (1.2),
where the constant factor is the best possible.
where the constant factor is the best possible; Krnić and Pečarić  also considered (1.11) in the general homogeneous kernel, but the best possible property of the constant factor was not proved by .
For in [10, inequality (37)], it reduces to the equivalent result of (3.1) in this paper.
In 2006-2007, some authors also studied the operator expressing of (1.3) and (1.4).
where the constant factor is the best possible. In particular, for being -degree homogeneous, inequalities (1.15) reduce to (1.3)-(1.4) (in the symmetric kernel). Yang  also considered (1.15) in the real space .
In this paper, by using the way of weight coefficient and the theory of operators, we define a new Hilbert-type operator and obtain its norm. As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernel of -degree is established; some particular cases are considered.
2. On a New Hilbert-Type Operator and the Norm
If is a measurable function, satisfying for then we call the homogeneous function of -degree.
For setting we find Hence, the following two words are equivalent: (a) is decreasing in and strictly decreasing in a subinterval of ; (b) for any , is decreasing in and strictly decreasing in a subinterval of . The following two words are also equivalent: is decreasing in and strictly decreasing in a subinterval of ; for any , is decreasing in and strictly decreasing in a subinterval of .
Setting by the assumption, we obtain (ii) Setting and in the integrals and respectively, in view of (i), we still find that
For we set and Define the real space as and then we may also define the spaces and
As the assumption of Lemma 2.2, for setting , if and are decreasing in and strictly decreasing in a subinterval of , then
In view of Lemma 2.3, and then exists. If there exists such that for any then is bounded and Hence by (2.4), we find and is bounded.
As the assumption of Lemma 2.3, it follows
Hence is the best value of (2.7). We conform that is the best value of (2.4). Otherwise, we can get a contradiction by (2.6) that the constant factor in (2.7) is not the best possible. It follows that
3. An Extended Basic Theorem on Hilbert-Type Inequalities
Still setting , and we have the following theorem.
where the constant factors and are the best possible.
In view of (2.7) and (2.4), we have (3.1) and (3.2). Based on Theorem 2.4, it follows that the constant factors in (3.1) and (3.2) are the best possible.
and we have (3.2). Hence (3.1) and (3.2) are equivalent.
For (3.1) and (3.2) reduce, respectively, to (1.6) and (1.7). Hence, Theorem 3.1 is an extension of Theorem A.
Replacing the condition " and are decreasing in and strictly decreasing in a subinterval of " by "for and are decreasing in and strictly decreasing in a subinterval of ," the theorem is still valid. Then in particular,
and then it deduces to (1.11);
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