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Abstract. In this paper, we introduce a new type of limit process to evaluate the modular-type operator norm of an integral operator. This leads us to get multidimensional extensions of Pólya-Knopp-type inequalities with general measures. Our results not only extend Levin-Cochran-Leetype inequalities from n = 1 to general n , but also improve the estimates given there. Moreover, they generalize Carleson's result, which is involved in the proof of Carleman's inequality. Besides these, the Pólya-Knopp-type inequalities for the cases of Laplace transform and generalized Riemann-Liouville operators are derived. For the lower bounds, a parallel theory to the above is also established. Mathematics subject classification (2010): 47A30, 26D10, 26D15.

In this paper, the modular-type operator norm of the general geometric mean operator over spherical cones is investigated. We give two applications of a new limit process, introduced by the present authors, to the establishment of Pólya-Knopp-type inequalities. We not only partially generalize the sufficient parts of Persson-Stepanov's and Wedestig's results, but we also provide new proofs to these results. MSC: 47A30; 26D10; 26D15