For the Hilbert type multiple integral inequality $$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$ with a nonhomogeneous kernel $K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })$ ( $\lambda _{1}\lambda _{2}> 0$ ), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, $\lambda _{1}$ , $\lambda _{2}$ , α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

中文翻译：

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一类具有非齐次核的最优Hilbert型多重积分不等式的有效性的条件

对于Hilbert型多重积分不等式$$ \ int _ {\ mathbb {R} _ {+} ^ {n}} \ int _ {\ mathbb {R} _ {+} ^ {m}} K \ bigl（\垂直x \ Vert _ {m，\ rho}，\ Vert y \ Vert _ {n，\ rho} \ bigr）f（x）g（y）\，\ mathrm {d} x \，\ mathrm {d} y \ leq M \ Vert f \ Vert _ {p，\ alpha} \ Vert g \ Vert _ {q，\ beta} $$，具有不均一的内核$ K（\ | x \ | _ {m，\ rho}， \ | y \ | _ {n，\ rho}）= G（\ | x \ | ^ {\ lambda _ {1}} _ {m，\ rho} / \ | y \ | ^ {\ lambda _ {2 }} _ {n，\ rho}）$（$ \ lambda _ {1} \ lambda _ {2}> 0 $），在本文中，通过使用权重函数方法，确定了参数p，q的充要条件，$ \ lambda _ {1} $，$ \ lambda _ {2} $，α，β，m和n应该满足，以使某个常数M的不等式成立，并建立最佳常数因子的表达式也获得了。最后，还考虑了它们在算子有界和算子范数中的应用，