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Zbl 0563.46024
Caffarelli, L.; Kohn, R.; Nirenberg, Louis
First order interpolation inequalities with weights. (English)
[J] Compos. Math. 53, 259-275 (1984). ISSN 0010-437X; ISSN 1570-5846/e

The authors prove a necessary and sufficient condition for there to exist a constant C such that for each $u\in C\sb 0\sp{\infty}$ $(R\sp n)$, $$ \Vert \vert x\vert\sp{\gamma}u\Vert\sb{L\sp r}\le C\Vert \vert x\vert\sp{\alpha}\vert Du\vert \Vert\sp a\sb{L\sp p}\Vert \vert x\vert\sp{\beta}u\Vert\sp{1-a}\sb{L\sp q}, $$ where $\alpha$, $\beta$, $\gamma$, a, r, p, q, and n are fixed real numbers satisfying a number of specified relationships. Special cases of this inequality have appeared in a number of papers, including a previous paper of the authors [Comm. Pure Appl. Math. 35, 771-831 (1982; Zbl 0509.35067)] and a paper of {\it B. Muckenhoupt} and {\it R. Wheeden} [Trans. Am. Math. Soc. 192, 261-274 (1974; Zbl 0289.26010)]. The proof is lengthy but elementary, and consists of verifying a large number of cases.
[P.Lappan]

MSC 2000:: *46E35 Sobolev spaces and generalizations
26D10 Inequalities involving derivatives, diff. and integral operators
46M35 Abstract interpolation of topological linear spaces
26D20 Analytical inequalities involving real functions

Citations: Zbl 0509.35067; Zbl 0289.26010

Cited in: Zbl 1249.35059 Zbl 1209.46017 Zbl 1177.35062 Zbl 1173.46015 Zbl 1156.35029 Zbl 1174.26324 Zbl 1143.53036 Zbl 1066.35034 Zbl 1037.26014 Zbl 0960.35039

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