Hardy's inequality and related topics

Abstract: The talk will deal mainly with the inequality $$ \left(\int_a^b|u(t)|^qw_0(t)\,dt\right)^{1/q}\le C\left(\int_a^b|u^{(k)}(t)|^pw_k(t)\,dt\right)^{1/p} \tag"($\ast$)" $$ for a natural number $k>1$, $0<q<\infty$, $1<p<\infty$. In dependence on the class of functions $u$ on which ($\ast$) is considered, necessary and sufficient conditions for $p$, $q$, $w_0$, $w_k$ will be derived which ensure the validity of ($\ast$). Further, the problem of compactness of the imbedding expressed by ($\ast$) will be dealt with, and the case of fractional order Hardy inequalities (i.e., inequalities of the type ($\ast$) with non-integer $k$) will be investigated. Some open problems will be formulated.

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