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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 Reviews
Cited 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
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\textit{L. Caffarelli} et al., Compos. Math. 53, 259--275 (1984; Zbl 0563.46024)
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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
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