J. Eur. Math. Soc. 11, 1105–1139 c  European Mathematical Society 2009 A. Cianchi · N. Fusco · F. Maggi · A. Pratelli The sharp Sobolev inequality in quantitative form Received November 4, 2007 and in revised form January 21, 2009 Abstract. A quantitative version of the sharp Sobolev inequality in W 1,p (R n ),1 <p<n, is established with a remainder term involving the distance from the family of extremals. 1. Introduction and main result A sharp form of the standard Sobolev inequality in R n , n ≥ 2, tells us that if 1 <p<n and p ∗ = np/(n − p), then S(p,n)‖f ‖ L p ∗ (R n ) ≤ ‖∇f ‖ L p (R n ) (1.1) for every function f from the homogeneous Sobolev space W 1,p (R n ) of functions f ∈ L p ∗ (R n ) such that ∇f ∈ L p (R n ). Here S(p,n) = √ πn 1/p  n − p p − 1  (p−1)/p  Ŵ(n/p)Ŵ(1 + n − n/p) Ŵ(1 + n/2)Ŵ(n)  1/n is the best possible constant in (1.1), and ‖∇f ‖ L p (R n ) stands for the L p (R n ) norm of the length |∇f | of the gradient ([Au, Ta]). A family of extremals in (1.1) is given by the functions g a,b,x 0 : R n → [0, ∞) defined as g a,b,x 0 (x) = a (1 + b|x − x 0 | p ′ ) (n−p)/p for x ∈ R n (1.2) for some a  = 0, b> 0, x 0 ∈ R n . Here, p ′ = p/(p − 1), the H¨ older conjugate of p. In fact, as pointed out by the recent contribution [CNV], functions having the form (1.2) are the only ones attaining equality in (1.1) for every p ∈ (1,n). Incidentally, note that, A. Cianchi: Dipartimento di Matematica e Applicazioni per l’Architettura, Piazza Ghiberti 27, 50122 Firenze, Italy; e-mail: cianchi@unifi.it N. Fusco: Dipartimento di Matematica ed Applicazioni, via Cintia, 80126 Napoli, Italy; e-mail: nicola.fusco@unina.it F. Maggi: Dipartimento di Matematica, viale Morgagni 67/A, 50134 Firenze, Italy; e-mail: maggi@math.unifi.it A. Pratelli: Dipartimento di Matematica, via Ferrata 1, 27100 Pavia, Italy; e-mail: aldo.pratelli@unipv.it