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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 uC 0 (R n ),

|x| γ u L r C|x| α |Du| L p a |x| β u L q 1-a ,

where α, β, γ, 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

MSC:
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