On Sobolev-Višik-Dubinskiĭ type inequalities (Q2724908)

The author establishes some new results related to some Sobolev-Vishik-Dubinskij type inequalities. Let \(a_i,b_i\in R= (-\infty,\infty)\), \(a_i< b_i\) for \(i= 1,2,\dots, n\), \(B= \prod^n_{i=1} [a_i,b_i]\), and \(p\geq 0\), \(q\geq 1\), \(m\geq 0\), \(r\geq 1\), \(\mu> 0\) are real constants. Denote by \(S\) the set of all continuous functions \(u(x)= u(x_1,\dots, x_n)\in C(B, R)\), with \(u(x)|_{x_i= a_i}= u(x)|_{x_i= b_i}\), \(i= 1,\dots, n\), and suppose that its first-order partial derivatives exist. The inequalities obtained can be briefly restated as follows: For \(u\in S\) we have NEWLINE\[NEWLINE\int_B|u(x)|^{r(p+ q)} V(x)^m dx\leq \min\Biggl[L^q \int_B|u(x)|^{rp} V(x)^{q+m} dx, L^{p+q} \int_B V(x)^{p+ q+m} dx\Biggr],\tag{\(*\)}NEWLINE\]NEWLINE where \(V(x):= (\sum^n_{i=1}|{\delta\over\delta x_i} u(x)|^\mu)^{r/\mu}\) and NEWLINE\[NEWLINEL= (p+ q+ m)^r n^{-\min(1,r/\mu)} \int^1_0 [t^{1-r}+ (1- t)^{1/r}]^{-1} dt\cdot\prod^n_{i=1} (b_i- a_i)^{r/n}.NEWLINE\]NEWLINE We note here that above inequality \((*)\) contains the inequalities (6) and (7) given in Theorem 2 and its special case when \(m= 0\) yields a brief statement of the Theorem 1. The method of proof used in the paper is new and elementary.