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On integral inequalities of the Sobolev type. (English) Zbl 0826.26007

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.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
39B72 Systems of functional equations and inequalities
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