By means of the techniques of the real analysis and the weight functions, a few equivalent conditions of a Hilbert-type integral inequality with the general nonhomogeneous kernel in the whole plane are obtained. The constant factor is proved to be the best possible. As applications, a few equivalent conditions of a Hilbert-type integral inequality with the general homogeneous kernel in the whole plane are deduced. We also consider the operator expressions, a few particular cases and some examples.

Key words: Hilbert-type integral inequality; weight function; equivalent form; operator; norm

2000 Mathematics Subject Classification. 26D15