Abstract
We first establish the Muckenhoupt-type estimation for the best constant $C$ associated with the followingmultidimensional modular inequality over a spherical cone:$$\biggr(\int_E\biggl \{\Phi\left(\int_{\tildeS_x}k(x,t)f(t)d\sigma(t)\right)\biggr \}^qd\mu\biggr)^{1/q}\leC\biggr(\int_E \biggl \{\Phi (f(x))\biggr \}^pd\nu\biggr)^{1/p},$$where $f\in L^p_{\Phi}(d\nu)$ and $1\le p, q\le \infty$. Here, $\Phi$ is a nonnegative convex function. Similar results are also derived for the complementary integral operator. Next, we introduce a new limiting procedure to get the P\'olya-Knopp type inequality from the original Hardy-type inequality. The estimation derived by this procedure improves the corresponding estimations in many cases. Last, we consider the multidimensional extensions of Manakov's result. Under some additional conditions on $\nu$, the estimate of the best constant can be improved. We also extend these results to their P\'olya-Knopp forms. Our results in thesis generalize many well-known recent works.