# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.