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Interpolation inequalities in Besov spaces. (English) Zbl 1022.46018
In the paper under review, the authors prove an interpolation inequality which extends various well-known inequalities in special cases. More specifically, they show that for $\lambda,$ $\mu$, $p,q,r$, $\theta$ satisfying $\lambda, \mu \in \Bbb R,$ $1\leq p,q\leq r\leq \infty$, $0<\theta<1$; $\lambda>\frac {n}{p}-\frac{n}{r},$ $\mu<\frac{n}{q}-\frac{n}{r},$ $\theta(\lambda-\frac{n}{p}+\frac{n}{r})+(1-\theta)(\mu-\frac{n}{q}+\frac{n}{r})=0$, there exists a constant $C>0$ such that $$||f;\overset \cdot \to B^{0}_{r,1}||\leq C||f;\overset\cdot\to B^{\lambda}_{p,\infty}||^{\theta}||f;\overset\cdot\to B^{\mu}_{q,\infty}||^{1-\theta} $$ for all $f\in \overset\cdot\to B^{\lambda}_{p,\infty}\cap \overset\cdot \to B^{\mu}_{q,\infty}$.

46B70Interpolation between normed linear spaces
46M35Abstract interpolation of topological linear spaces
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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