Affiliations: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warszawa, Poland. E-mails: [email protected], [email protected]
Note: [] Corresponding author: Agnieszka Kałamajska, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02–097 Warszawa, Poland. E-mail: [email protected].
Abstract: We derive the inequality ∫R|f′(x)|ph(f(x)) dx≤(\sqrt{p−1})p∫R(\sqrt{|f″(x)𝒯h(f(x))|})ph(f(x)) dx, where f belongs locally to the Sobolev space W2,1 and f′ has bounded support. Here h(·) is a given function and 𝒯h(·) is its given transform, it is independent of p. In case when h≡1 we retrieve the well-known inequality: ∫R|f′(x)|p dx≤(\sqrt{p−1})p∫R(\sqrt{|f″(x)f(x)|})p dx. Our inequalities have a form similar to the classical second-order Opial inequalities. They also extend certain class of inequalities due to Mazya, used to obtain second-order isoperimetric inequalities and capacitary estimates. We apply them to obtain new a priori estimates for nonlinear eigenvalue problems.
