The best constants for multidimensional modular inequalities over spherical cones

Abstract

In this paper, we establish the Muckenhoupt-type estimation for the best constant associated with the following multidimensional modular inequality over a spherical cone:where and . Similar results are also derived for the complementary integral operator. Our results provide the -dimensional modular forms of the works of Andersen and Heinig. As consequences of our results, we give the -dimensional weighted extensions of Levinson modular inequality, extensions of Stepanov and Heinig results, generalizations of the Hardy–Knopp-type inequalities, and those for the Riemann–Liouville operator and the Weyl fractional operator. We also point out that our estimates are better than those given in the works of Drabek, Heinig, Kunfer, and Sinnamon.

Keywords:

• operator norm
• integral operator
• multidimensional modular inequality
• Hardy–Knopp-type inequalities

AMS Subject Classifications:

• 26D10
• 26D15

Acknowledgments

The first author was supported in part by the National Science Council, Taipei, ROC, under Grant NSC 99-2115-M-364-001-MY3. The third author was supported in part by the National Science Council, Taipei, ROC, under Grant NSC 100-2115-M-214-004.