The parameter conditions for the existence of the Hilbert-type multiple integral inequality and its best constant factor

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.