Caffarelli, L.; Kohn, R.; Nirenberg, Louis First order interpolation inequalities with weights. (English) Zbl 0563.46024 Compos. Math. 53, 259-275 (1984). 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 Cited in 15 ReviewsCited in 501 Documents 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 26D20 Other analytical inequalities Citations:Zbl 0509.35067; Zbl 0289.26010 PDF BibTeX XML Cite \textit{L. Caffarelli} et al., Compos. Math. 53, 259--275 (1984; Zbl 0563.46024) Full Text: Numdam EuDML OpenURL References: [1] J. Scott Bradley : Hardy inequalities with mixed norms . Canad. Math Bull 21 (1978) 405-408. · Zbl 0402.26006 [2] L. Caffarelli , R. Kohn and L. Nirenberg : Partial regularity of suitable weak solutions of the Navier-Stokes equations . Comm. Pure Appl. Math, 35 (1982) 771-831. · Zbl 0509.35067 [3] E. Gagliardo : Ulteriori proprietà di alcune classi di funzioni in più variabili . Ricerche di Mat. Napoli 8 (1959) 24-51. · Zbl 0199.44701 [4] G.H. Hardy , J.E. Littlewood and G. Polya : Inequalities . Cambridge: Cambridge University Press (1952). · JFM 60.0169.01 [5] C.S. Lin : Interpolation inequalities with weighted norms , to appear. [6] B. Muckenhoupt : Hardy’s inequality with weights . Studia Math. 34 (1972) 31-38. · Zbl 0236.26015 [7] B. Muckenhoupt and R. Wheeden : Weighted norm inequalities for fractional integrals . Trans. Amer. Math. Soc. 192 (1974) 261-274. · Zbl 0289.26010 [8] L. Nirenberg : On elliptic partial differential equations . Ann. di Pisa 9 (1959) 115-162. · Zbl 0088.07601 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.