STUDIA MATHEMATICA 159 (2) (2003)A new characterization of the Sobolev spacebyPiotr Hajłasz (Warszawa)Abstract. The purpose of this paper is to provide a new characterization of theSobolev space W1,1(Rn). We also show a new proof of the characterization of the Sobolevspace W1,p(Rn), 1 ≤ p < ∞, in terms of Poincar´e inequalities.The Sobolev space W1,p(Rn), 1 ≤ p < ∞, consists of functions u ∈Lp(Rn) such that |∇u| ∈ Lp(Rn). It is a Banach space with respect to thenormkuk1,p= kukp+ k∇ukp.Let us point out that from the point of view of Banach spaces the structureof various Sobolev type spaces, with the particular emphasis on W1,1, hasbeen investigated by A. Pe lczy´nski, M. Wojciechowski and others; see e.g.[4], [40]–[43] and references therein.The purpose of this paper is to provide a new characterization of theSobolev space W1,1(Rn). Here, however, we emphasize future applicationsin geometric analysis and analysis on metric spaces rather than the theory ofBanach spaces. Actually one of the reasons for finding new characterizationsof the Sobolev space is the development of analysis on metric spaces; see e.g.[1], [2], [5], [6], [12], [13], [15]–[19], [21], [23], [24], [28]–[31], [38], [47]–[49],[51]. More references will be given later. In order to define a Sobolev typespace on a metric-measure space we need a characterization of the spaceW1,p(Rn) that does not involve derivatives. One such characterization isgiven in the following result.Theorem 1 ([19]). u ∈ W1,p(Rn), 1 < p < ∞, if and only if u ∈ Lp(Rn)and there exists 0 ≤ g ∈ Lp(Rn) such that|u(x) − u(y)| ≤ |x − y|(g(x) + g(y)) a.e.(1)2000 Mathematics Subject Classification: Primary 46E35.This work was supported by the KBN grant 2 PO3A 028 22. The research was com-pleted during the stay of the author in Department of Mathematics at University ofMichigan. The author wishes to thank the University for the support and hospitality.[263]264 P. HajłaszMoreoverk∇ukp≈ infgkgkp,where the infimum is taken over the class of functions g satisfying (1).Inequality (1) holds a.e. in the sense that there is a set E ⊂ Rnofmeasure zero such that (1) holds for all x, y ∈ Rn\ E. Writing A ≈ B wemean that there is a constant C ≥ 1 such that C−1B ≤ A ≤ CB.Let us note that yet another characterization of the Sobolev space hasbeen obtained recently in [9] and [10].The above theorem was a point of departure in [19] for the definition ofa Sobolev space on an arbitrary metric space equipped with a locally finiteBorel measure. For further results involving this approach see e.g. [3], [8],[13], [15], [16], [18], [20]–[22], [24]–[30], [32]–[37], [39], [44], [45], [49], [52], [53].If u ∈ W1,p(Rn), 1 ≤ p < ∞, then we have an elementary inequality ([7],[19])|u(x)−u(y)| ≤ C(n)|x−y|(M2|x−y||∇u|(x)+M2|x−y||∇u|(y)) a.e.(2)whereMRg(x) = supr<RB(x,r)|g(z)| dzis the restricted Hardy–Littlewood maximal function. Here and in whatfollows the integral average of a function u over a set E is denoted byuE=Eu dx =1|E|Eu dx,where |E| denotes Lebesgue measure of E. Moreover C will always standfor a general constant that can change its value even in the same string ofestimates. Writing C = C(n) we will emphasize that the constant dependson n only.If we take R = ∞ in the definition of MR, then we obtain the classicalHardy–Littlewood maximal functionMu(x) = supr>0B(x,r)|u| .Hence it follows from inequality (2) that|u(x) − u(y)| ≤ C|x − y|(M|∇u|(x) + M|∇u|(y)) a.e.This and the boundedness of the maximal function in Lpfor p > 1 (see [50])imply (1) with g = CM|∇u| ∈ Lp(Rn). The implication in the oppositedirection in Theorem 1 follows from the lemma.Lemma 2 ([20, Proposition 1]; cf. [28, Remark 5.13]). If u ∈ L1loc(Rn)and 0 ≤ g ∈ L1loc(Rn) satisfy inequality (1) a.e., then ∇u ∈ L1loc(Rn) inCharacterization of Sobolev space 265the weak sense and|∇u| ≤ C(n)g a.e.(3)This lemma is relatively easy and its proof is based on the observationthat (1) implies absolute continuity of u on almost all lines parallel to co-ordinate axes. If we know in addition that g ∈ Lp(Rn), then inequality (3)implies that |∇u| ∈ Lp(Rn), which completes the proof of Theorem 1.Observe that the only place in the proof of Theorem 1 where the as-sumption p > 1 was employed was the application of the boundedness ofthe maximal function in Lp. It turns out that the assumption p > 1 is es-sential because Theorem 1 does not hold for p = 1. This follows from thenext example.Example 3 ([20]). Let Ω = (−1/2, 1/2) and u(x) = −x/(|x| log |x|).Then u ∈ W1,1(Ω) because u0(x) = |x|−1(log |x|)−2∈ L1(Ω). Suppose nowthat there exists 0 ≤ g ∈ L1(−1/2, 1/2) such that (1) holds a.e. Then fora.e. 0 < x < 1/2 we have |u(x) − u(−x)| ≤ 2x(g(x) + g(−x)) and hence−2log x≤ 2x(g(x) + g(−x)),which, in turn, yields1/2−1/2g(x) dx =1/20(g(x) + g(−x)) dx ≥1/20−dxx log x= ∞.This contradicts integrability of g. The function u is defined on the interval(−1/2, 1/2) only, but one can extend it to a function in W1,1(R) to fit intothe setting of Theorem 1.The main result of the present paper is the following characterization ofW1,1(Rn).Theorem 4. u ∈ W1,1(Rn) if and only if u ∈ L1(Rn) and there exist0 ≤ g ∈ L1(Rn) and σ ≥ 1 such that|u(x) − u(y)| ≤ |x − y|(Mσ|x−y|g(x) + Mσ|x−y|g(y)) a.e.(4)Moreover if (4) holds, then |∇u| ≤ C(n, σ)g a.e.The implication from left to right follows from the elementary inequalityat (2). It turns out, however, that the implication from right to left is muchmore difficult than the corresponding one in Theorem 1.The proof that inequality (4) implies u ∈ W1,1(Rn) is split into twosteps. In the first step we prove that (4) implies the family of Poincar´e typeinequalitiesB|u − uB| ≤ Cr3σBg(5)266 P. Hajłaszfor every ball B of any radius r. Here and in what follows, 3σB denotes theball concentric with B and with radius 3σ times that of B. Observe thatinequality (5) with 3σB replaced by B would readily follow from (1) uponintegration over x, y ∈ B. In our situation, however, we cannot integrate(4) because the maximal function of an L1function need not be integrable.This is the main difficulty in the proof and, actually, this first step is themain new ingredient in the proof.In the second step we show that the family of inequalities (5) on everyball B imply that u ∈ W1,1(Rn) with |∇u|