On maximal overdetermined Hardy's inequality of second order on a finite interval

Maria Nasyrova; Vladimir D. Stepanov

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The Hardy inequality $\int_{Ω} {| u (x) |}^{p} d {(x)}^{- p} \ddot{x} \leq c \int_{Ω} {| \nabla u (x) |}^{p} \ddot{x}$ with $d (x) = dist (x, \partial Ω)$ holds for $u \in C_{0}^{\infty} (Ω)$ if $Ω \subset ℝ^{n}$ is an open set with a sufficiently smooth boundary and if $1 < p < \infty$ . P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p = 1$ .