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 \(u\in C_ 0^{\infty}\) \((R^ n)\), \[ \| | x|^{\gamma}u\|_{L^ r}\leq C\| | x|^{\alpha}| Du| \|^ a_{L^ p}\| | x|^{\beta}u\|^{1-a}_{L^ q}, \] 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 B. Muckenhoupt and 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


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
26D20 Other analytical inequalities
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