An extended Hilbert’s integral inequality in the whole plane with parameters

By introducing independent parameters and interval variables, applying the weight functions and the technique of real analysis, an extended Hilbert’s integral inequality in the whole plane with parameters and a best possible constant factor is provided. The equivalent forms, the reverses, and the related homogeneous forms with particular parameters are considered. Meanwhile, an extended Hilbert’s integral operator in the whole plane is defined, and the operator expressions for the equivalent inequalities are obtained.

In this paper, by introducing independent parameters and interval variables, applying the weight functions and the technique of real analysis, a Hilbert-type integral inequality in the whole plane with parameters and a best possible constant factor is provided as follows: (μ, σ > 0, μ + σ = λ), which is an extension of (4). The more general form of (6) with parameters, the equivalent inequalities, the reverses, and the related homogeneous form with the particular parameter are considered. Meanwhile, an extended Hilbert's integral operator in the whole plane is defined, and the operator expressions for the equivalent inequalities are obtained.
Proof (i) For p > 1, by Hölder's inequality with weight [15] and (7), when y = 0, we find We prove that (13) takes the form of strict inequality. Otherwise, there exists y = 0 such that (13) takes the form of equality. Then there exist constants A and B such that they are not all zero, and [15] If A = 0, then B = 0, which is impossible. We suppose that A = 0, namely which contradicts the fact that 0 < Then by (11) and Fubini's theorem [16], we find In view of (10) and (11), we have (12).
The theorem is proved.
On the other hand, suppose that (15) is valid. We set By (14) and the assumptions, we find J < ∞. If J = 0, then (12) is trivially valid; if J > 0, then by (15) Hence, we have (12), which is equivalent to (15).
For n ∈ N = {1, 2, . . .}, n > 1 qμ , we define the sets E δ := {x ∈ R; |x| δ ≥ 1}, and the following functions: Then we obtain that In the same way, we still find that Hence, we obtaiñ If there exists a constant k ≤ K(σ ) such that (15) is valid when replacing K(σ ) by k, then in particular, we have In view of the above results, it follows that For n → ∞, we find namely K(σ ) ≤ k. Hence, k = K(σ ) is the best possible constant factor of (15).
The constant factor K(σ ) in (12) is also the best possible. Otherwise, we can conclude a contradiction by (18) that the constant factor in (17) is not the best possible.
The theorem is proved.

Theorem 3
With regards to the assumptions of Theorem 2, replacing p > 1 by 0 < p < 1, we have the equivalent reverses of (12) and (15) with the best possible constant factor K(σ ).
Proof We only prove that the constant factor K(σ ) in the reverse of (15) is the best possible, and omit the others. If there exists a constant k ≥ K(σ ) such that the reverse of (15) is valid when replacing K(σ ) by k, then in particular, for n ∈ N = {1, 2, . . .}, n > 1 |q|σ , we have For n → ∞, we obtain that k ≤ K(σ ). Hence, k = K(σ ) is the best possible constant factor of the reverse of (15). The theorem is proved.
In view of (20), the operator T is bounded with Since by Theorem 2 the constant factor in (20) is the best possible, we have If we define the normal inner product of Tf and g as follows: then we can rewrite (15) and (12) as the following equivalent operator expressions: (Tf , g) < T · f p,ϕ g q,ψ , Tf p,ψ 1-p < T · f p,ϕ .

Conclusions
In this paper, by introducing independent parameters and interval variables, applying the weight functions and the technique of real analysis, an extended Hilbert's integral inequality in the whole plane with parameters and a best possible constant factor is provided in Theorem 2. The equivalent forms, the reverses, and the related homogeneous forms with particular parameters are considered. An extended Hilbert's integral operator in the whole plane is defined, and the operator expressions for the equivalent inequalities are obtained. The method of weight functions is very important, which helps us to prove the equivalent inequalities with the best possible constant factor. The lemmas and theorems provide an extensive account of this type of inequalities.

Funding
This work is supported by the National Natural Science Foundation (No. 61772140) and Science and Technology Planning Project of Guangzhou City (No. 201707010229). We are grateful for this help.