On integral inequalities of the Sobolev type
The main result states that for a rectangle \(\Omega= \prod^n_{i= 1} (a_i, b_i)\), for any functions \(f^\alpha\in C^1_0(\Omega)\), \(\alpha= 1,\dots, m\), and for any numbers \(p_\alpha\geq 1\), \(q_\alpha> 0\) with \(\sum p_\alpha/q_\alpha= 1\) the inequality \[ \int_\Omega \prod_h\alpha|f^\alpha|^{q_\alpha}\leq {M\over 2\sqrt n} \sum_\alpha \Biggl( \int_\Omega |f^\alpha|^{2(p_\alpha- 1)}\Biggr)^{1/2} \Biggl( \int_\Omega |\nabla f^\alpha|^2\Biggr)^{1/2}, \] holds with \(M= \max\{b_i- a_i: 1\leq i\leq n\}\). The elementary proof is based on the one-dimensional identity expressing the function by means of its integral of the derivative and on some analogues of the quadratic mean -- arithmetic mean -- geometric mean inequality.
scientific article; zbMATH DE number 729151
scientific article