Preface

One hundred years ago, in 1908, H. Wely published the well known Hilbert’s inequality. In 1925, G. H. Hardy gave its best extension by introducing one pair of conjugate exponents (p, q), named in Hardy-Hilbert’s inequality. The Hilbert-type inequalities are a more wide class of analysis inequalities which are with the bilinear kernels, including Hardy-Hilbert’s inequality as the particular case. These inequalities are important in analysis and its applications. After extensive efforts of mathematicians about one hundred years, the theory of Hilbert-type inequalities has now come into being. This book is a monograph about the theory of Hilbert-type integral inequalities with the homogeneous kernels of real number-degree and its applications. Using the methods of Real Analysis, Functional Analysis and Operator Theory, and following the way of weight functions and real analysis, the author introduces an independent parameter and two pairs of conjugate exponents to establish a number of Hilbert-type integral inequalities with the homogeneous kernels of real number-degree and the best constant factors, including some multiple integral inequalities and multivariable integral inequalities. The equivalent forms and the reverses with the best constant factors are also considered. As applications, the author also considers some Hilbert-type integral inequalities with the non-homogeneous kernels, the Hardy-type integral inequalities as the particular kernels and a large number of particular examples.

For reading and understanding this book, readers should hold the basic knowledge of real analysis and functional analysis. This book is appropriate for those who are interested in the fields of analysis inequalities and real analysis. The author expects this book can help many readers to make good progresses in research for Hilbert-type integral inequalities and their applications.