Abstract
By means of the weight function, the following results are given. The Hilbert-type multiple integral inequality with the \(\lambda \)-order homogeneous kernel \(\int _{R_{+}^{n}}\int _{R_{+}^{m}}K(\left\| x\right\| _{m,\rho },\left\| y\right\| _{n,\rho })f(x)g(y)\mathrm{d}x\mathrm{d}y\le M\left\| f\right\| _{p,\alpha }\left\| g\right\| _{q,\beta }\) is true if and only if \(\frac{\alpha +m}{p}+\frac{\beta +n}{q}=\lambda +m+n\), and the expression of the best possible constant factor is obtained. Furthermore, its application in the operator theory is discussed.
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The authors would like to thank the referees for their valuable comments. This work was supported by the National Natural Science Foundation (No. 61772140).
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Communicated by Kasso Okoudjou.
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Hong, Y., Huang, Q. & Chen, Q. The parameter conditions for the existence of the Hilbert-type multiple integral inequality and its best constant factor. Ann. Funct. Anal. 12, 7 (2021). https://doi.org/10.1007/s43034-020-00087-5
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DOI: https://doi.org/10.1007/s43034-020-00087-5
Keywords
- Hilbert-type multiple integral inequality
- Homogeneous kernel
- Parameter condition
- Bounded operator
- Operator norm