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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
##### Keywords:
interpolation inequality; Lebesgue norm; Hölder norm
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##### References:
  L. Nirenberg, On elliptic partial differential equations.Ann. Scuola Norm. Sup. Pisa 13(1959), 115–162. · Zbl 0088.07601  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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