Weighted multidimensional integral inequalities and applications

Abstract: In this thesis some new weighted integral inequalities for monotone functions in higher dimensions are proved. These results extend previous results in one dimension, and also those in higher dimensions for particular choices of the weights (power weights, etc.). All inequalities are sharp. A new duality principle (of Sawyer type) over the cone of multidimensional monotone functions is proved and applied. Weighted Chebyshev type inequalities for monotone functions and modular inequalities in higher dimensions are proved. A new type of weighted function spaces are introduced. In particular these spaces generalize the classical Lebesgue spaces. The weights such that they become quasi-Banach spaces are completely characterized. A multidimensional multiplicative inequality (of Carlson type) for weighted Lebesgue spaces with homogeneous weights is proved and applied. The inequality is sharp and all cases of equality are pointed out.