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An interpolation inequality involving Hölder norms. (English) Zbl 0855.26007
The well-known inequality proved by L. Nirenberg [Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 13, 115-162 (1959; Zbl 0088.07601)] states that all functions \(u\in C^\infty_0(\mathbb{R}^N)\) satisfy the inequality \(|\nabla^j u|_q\leq C|\nabla^m u|^a_p |u|^{1- a}_r\), where \(|\cdot |_s\) is the \(L^s\)-norm, the parameters satisfy the conditions \(0< j< m\), \(j/m\leq a< 1- j+ N/q= a(-m+ N/p)+ (1- a) N/r\) and \(C\) is a positive constant independent of \(u\). The authors prove an analogue of the inequality with the Hölder quotient \([u]_{H(\lambda)}= \sup_{x\neq y} |u(x)- u(y)|/|x- y|^\lambda\), \(0< \lambda< 1\), in place of \(|u|_r\) on the right-hand side. For \(N= 1\) the parameters satisfy the assumptions \(1< p< q< \infty\), \(0< \lambda\leq 1\) (\(0< \lambda< 1- 1/q\) if \(j =1\)), \(q> mp/j\), \(\lambda\leq (jq- mp)/(q- p)\), and \(- j+ 1/q= a(- m+ 1/p)+ (1- a)(- \lambda)\). In the multidimensional case, they are assumed to satisfy the more restrictive conditions \(1< p< q< \infty\), \(mp/j< q\leq (m- 1) p/(j- 1)\), \(\lambda= (jq- mp)/(j- 1)\), and \(a= p/q\).

MSC:
26D10 Inequalities involving derivatives and differential and integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:
[1] L. Nirenberg, On elliptic partial differential equations.Ann. Scuola Norm. Sup. Pisa 13(1959), 115–162. · Zbl 0088.07601
[2] R. C. Brown and D. B. Hinton, Sufficient conditions for weighted inequalities of sum form.J. Math. Anal. Appl. 112(1985), 563–578. · Zbl 0587.26011
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