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First order interpolation inequalities with weights. (English) Zbl 0563.46024
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.
Reviewer: P.Lappan

46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10Inequalities involving derivatives, differential and integral operators
46M35Abstract interpolation of topological linear spaces
26D20Analytical inequalities involving real functions
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