×

On the way of weight function and research for Hilbert’s type integral inequalities. (Chinese. English summary) Zbl 1114.26024

This paper summarizes the results in the research of Hilbert type inequalities generalized from the classic Hilbert inequality, \[ \int_0^\infty \int_0^\infty \frac {f(x)g(x)}{x+y} dx dy < \pi \left( \int_0^\infty f^2(x) dx \int_0^\infty g^2(x) dx \right)^{1/2}, \] mostly obtained in the recent decade, including the author’s own work. Constructing weight functions is the main technique used in the proofs of Hilbert type inequalities. The paper introduces the weight functions corresponding to some generalized Hilbert type inequalities. The paper also introduces two optimal generalizations of the Hilbert inequality with the \((p, q)\) indices pair, one is invertible and the other one is not. Generalizations of the Hilbert type inequalities in higher dimensional spaces are introduced and three corresponding open questions are listed. Finally the paper introduces a specific Hilbert type inequality involving the \((p, q)\) indices pair and a constant \(\lambda\) obtained by the author and lists three open questions about the inequality.

MSC:

26D15 Inequalities for sums, series and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDFBibTeX XMLCite