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Interpolation inequalities with weights. (English) Zbl 0635.46032
Summary: We use elementary methods to prove a sufficient and necessary condition for Sobolev interpolation inequalities with weight $| | x|^{\gamma}D\quad ju|_ r\leq C| | x|^{\alpha}D\quad mu| \quad a_{L\quad p}| | x|^{\beta}u|_{L\quad q}^{1-a},$ where p, $$\alpha$$, q, $$\beta$$, $$\gamma$$, r are real numbers, and $$p,q\geq 1,1/p+\alpha /n,1/q+\beta /n,1/r+\gamma /n>0$$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives and differential and integral operators 46M35 Abstract interpolation of topological vector spaces
##### Keywords:
Sobolev interpolation inequalities
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##### References:
 [1] Caffarelli L., Compositio Math. 53 pp 259– (1984) [2] Nirenberg L., Ann. di Pisa 9 pp 1515– (1962) [3] Friedman A., Partial different ialequations
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