A Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree \(-2\)
Assume that \(f(x)\) and \(g(x)\) are non-negative and both of them satisfy \(0<\int_0^\infty f^2(x)\,dx<\infty\). Then, Hilbert's integral inequality asserts that \[ \int_0^\infty \int_0^\infty \frac{f(x)g(y)}{x+y}\,dx\,dy< \pi\left(\int_0^\infty f^2(x)\,dx\int_0^\infty g^2(x)\,dx\right)^\frac12, \] where \(\pi\) is the best possible constant. The main object of the paper is to give a new Hilbert-type inequality in the whole plane with a homogeneous kernel of degree \(-2\) involving some parameters and a best constant factor.
scientific article; zbMATH DE number 5887677