On a Multiple Hilbert-Type Integral Inequality with the Symmetric Kernel

: We build a multiple Hilbert-type integral inequality with the symmetric kernel and involving an integral operator . For this objective, we introduce a norm , two pairs of conjugate exponents and , and two parameters. As applications, the equivalent form, the reverse forms, and some particular inequalities are given. We also prove that the constant factors in the new inequalities are all the best possible.