Abstract
We derive the inequality
with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space \({W^{2,1}(\mathbb{R})}\) and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and \({\tau_h(\cdot)}\) is its given transform independent of M. When M(λ) = λp and \({h \equiv 1}\) we retrieve the well-known inequality \({\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}\). We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).
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Kałamajska, A., Peszek, J. On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities. J. Fixed Point Theory Appl. 13, 271–290 (2013). https://doi.org/10.1007/s11784-013-0106-7
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DOI: https://doi.org/10.1007/s11784-013-0106-7