The parameter conditions for the existence of the Hilbert-type multiple integral inequality and its best constant factor
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00087-5 ORIGINAL PAPER
The parameter conditions for the existence of the Hilbert‑type multiple integral inequality and its best constant factor Yong Hong1 · Qiliang Huang2 · Qiang Chen3 Received: 23 April 2020 / Accepted: 23 July 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract By means of the weight function, the following results are given. The Hilbert-type multiple integral inequality with the 𝜆-order homogeneous kernel ∫Rn ∫Rm K(‖x‖m,𝜌 , ‖y‖n,𝜌 )f (x)g(y)dxdy ≤ M‖f ‖p,𝛼 ‖g‖q,𝛽 is true if and only if + + 𝛼+m + 𝛽+n = 𝜆 + m + n , and the expression of the best possible constant factor is p q obtained. Furthermore, its application in the operator theory is discussed. Keywords Hilbert-type multiple integral inequality · Homogeneous kernel · Parameter condition · Bounded operator · Operator norm Mathematics Subject Classification 26D10 · 26D15
Communicated by Kasso Okoudjou. * Qiliang Huang [email protected] Yong Hong [email protected] Qiang Chen [email protected] 1
Department of Mathematics, Guangdong Baiyun University, Guangzhou 510450, Guangdong, People’s Republic of China
2
Department of Mathematics, Guangdong University of Education, Guangzhou 510303, Guangdong, People’s Republic of China
3
Department of Computer Science, Guangdong University of Education, Guangzhou 510303, Guangdong, People’s Republic of China
Vol.:(0123456789)
Y. Hong et al.
1 Introduction For n ∈ 𝐍, x = (x1 , x2 , … , xn ), ‖x‖n,𝜌 = (x1𝜌 + x2𝜌 + ⋯ + xn𝜌 )1∕𝜌 (𝜌 > 0), 𝐑n+ = {(x1 , x2 , … , xn ) ∶ xi > 0, i = 1, 2, ⋯ , n} , we set
� Lpr (𝐑n+ )
= {f (x) ≥ 0 ∶ ‖f ‖p,r =
�1∕p
�𝐑n
‖x‖rn,𝜌 f p (x)dx
< +∞}.
+
In this paper, we research the conditions under which the following Hilbert-type multiple integral inequality is true:
�Rn �Rm +
K(‖x‖m,𝜌 , ‖y‖n,𝜌 )f (x)g(y)dxdy ≤ M‖f ‖p,𝛼 ‖g‖q,𝛽 .
(1.1)
+
There are two main concerns. When (1.1) is true, what conditions should its parameters satisfy? Furthermore, Are the conditions necessary and sufficient? We also consider its application in the operator theory. The authors mainly research the general theory of Hilbert-type inequality. Meanwhile, also discusses the Hilbert-Type inequality with various different kernels and its application. Especially in recent years, some good results have been obtained in the construction and high-dimensional extension of Hilbert-type inequality. Although there are lots of relevant research results (cf. [1, 3–11]), the issues discussed in this article are entirely new and of general significance. In this article, by using the methods and techniques of real analysis, the sufficient and necessary conditions for the Hilbert-type multiple integral inequality with the homogeneous kernel to be true, and the best possible constant factor are discussed in Theorem 3.1. Furthermore, its application in the operator theory is considered in Theorem 4.1. The method of real analysis is very important, which is the key to prove the equivalent inequalities with the best possible con
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