A Surprising Formula for Sobolev Norms

Home , Lp space, Seminorm, Sobolev space, Vitali covering lemma

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MATHEMATICS Z k(p, N ) := |e · ω|p dω, [1.6] The key to our proof is the following proposition, which when N −1 0 S γ = 1 and f = u gives [1.7] and thus yields the desired upper N −1 N bound for the p = 1 case of Theorem 1.1 in dimension N = 1. where S denotes the unit sphere in R , and e is any unit vector in N . R Proposition 2.1. There exists a universal constant C such that for all Some comments concerning the above results are in order. γ > 0 and all f ∈ Cc ( ), we have First, the validity of the upper bound in [1.3] when p = 1 is R quite remarkable and somewhat unexpected. In fact, a natural ZZ γ γ−1 5 strategy to establish this upper bound (such as the one pre- |x − y| dx dy ≤ C kf k 1 , L (R) sented in Remark 2.3 below) requires a strong type estimate E(f ,γ) γ for the maximal function (of the gradient of u), which holds when p > 1, but notoriously fails at the end-point p = 1. We where overcome this difficulty by applying the Vitali covering lemma Z x in a rather unconventional way which allows us to bypass the γ+1 E(f , γ) := (x, y) ∈ R × R: x 6= y, f ≥ |x − y| . obstruction commonly arising at p = 1 in this kind of situa- y tion. Thus, the hard core of the proof of the upper bound in [1.3] concerns the case p = 1. As it turns out, we can further- Proof: Since E(f , γ) ⊆ E(|f |, γ), without loss of generality more derive the case p > 1 from the case p = 1, at a crucial assume f is nonnegative. Let X be the collection of all nontrivial step of the argument. When p = 1 and N = 1, the upper bound closed intervals I ⊂ R such that in [1.3] amounts to the following “innocuous-looking” calculus inequality Z f ≥ |I |γ+1. [2.1] 2 2 2 I L (x, y) ∈ R : |u(x) − u(y)| ≥ |x − y| Z 0 (Here an interval is said to be nontrivial if it has positive length, ≤ C |u (t)|dt [1.7] and we used |I | to denote the length of the interval.) Then R ∞ [ for all u ∈ Cc (R), where C is a universal constant. Surprisingly, E(f , γ) ⊆ I × I . [2.2] this estimate seems to have gone unnoticed in the literature and I ∈X our proof is more involved than expected! Next, the lower bound in [1.3] is a consequence of Theorem 1/(γ+1) The lengths of all intervals in X are bounded by kf kL1 < ∞. 1.2. The proof of Theorem 1.2 involves original ideas, partially Hence we may apply the Vitali covering lemma and choose a inspired from techniques developed in ref. 3; actually, the con- subcollection Y of X , so that Y consists of a family of pairwise stant k(p, N ) in [1.6] already appeared in the BBM formula (ref. disjoint intervals J from X , and every I ∈ X is contained in 5J 3, theorem 1.2). for some J ∈ Y (see, e.g., ref. 11, claim in the proof of theorem The proof of the upper bound in [1.3] is presented in Section 1, section 1.5). It follows that 2. The proof of Theorem 1.2 is presented in Section 3. The assertions in Theorems 1.1 and 1.2, which are stated for [ ∞ N E(f , γ) ⊆ (5J ) × (5J ), [2.3] convenience when u ∈ Cc (R ), suggest that similar conclusions hold under minimal regularity assumptions on u and that the J∈Y W˙ 1,p 1 < p < ∞ BV˙ p = 1 Sobolev space , (respectively when ), where 5J is the interval with the same center as J but five times can be identified with the space of measurable functions u satisfy- p 2N p 2N the length. As a result, we see that ing supλ>0 λ L (Eλ) < ∞, or just lim supλ→∞ λ L (Eλ) < N ZZ ZZ ∞. One should also be able to replace R by domains Ω ⊂ γ−1 X γ−1 N , etc. We will return to this circle of ideas in a forth- |x − y| dx dy ≤ |x − y| dx dy R E(f ,γ) 5J×5J coming paper. J∈Y γ 10 · 5 X γ+1 = |J | . [2.4] 2. Proof of Theorem 1.1 γ(γ + 1) As already mentioned the lower-bound part is a consequence of J∈Y Theorem 1.2 whose proof is presented in Section 3: Indeed, if E λ (Here we used γ > 0 to integrate in x and y.) But for each J ∈ Y , is as in [1.4], then we have J ∈ X , so Z " #p γ+1 u(x) − u(y) p 2N |J | ≤ f . N = sup λ L (Eλ) J p +1 λ>0 |x − y| p N N M (R ×R ) Plugging this back into [2.4], we obtain p 2N ≥ lim λ L (Eλ) λ→∞ ZZ γ Z γ−1 5 X |x − y| dx dy ≤ C f and Holder’s¨ inequality gives E(f ,γ) γ J J∈Y [2.5] γ 1/p 5 k(p, N ) k(1, N ) k(1, N ) σ 1/p ≤ C kf kL1( ), ≥ = N −1 γ R 1 1 N 1− p σN −1 N σ N p N −1 the last inequality following from the disjointness of the different 1 1 ≥ k(1, N ) min , := c(N ), J ∈ Y . This completes the proof of Proposition 2.1. N σN −1 To prove the upper bound in Theorem 1.1 when N > 1 or N −1 where σN −1 denotes the surface area of S . Therefore, we p > 1, Proposition 2.1 still proves to be useful. Via the method concentrate here on the upper bound. of rotation, it implies the following proposition:

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MATHEMATICS ( ) |u(x) − u(y)| Ar ≤ δ|∇u(x) · ω| and λr N /p ≤ (1 − δ)|∇u(x) · ω| (x, y) ∈ N × N : x 6= y, ≥ λ R R N +1 |x − y| p imply that (x, y) ∈ Eλ. Thus, we can take R to be deﬁned by n N N ⊆ (x, y) ∈ R × R : N p N δ N (1 − δ) p N −1 o R(x, ω, λ) := min |∇u(x) · ω| , |∇u(x) · ω| . |x − y| p ≤ C λ (M|∇u|(x) + M|∇u|(y)) AN λp

n N N N −1 o ⊆ (x, y) ∈ R × R : |x − y| p ≤ 2 C λ M|∇u|(x) From [3.3] we have

n N N N −1 o ∪ (x, y) ∈ × : |x − y| p ≤ 2 C λ M|∇u|(y) ( R R 1 ZZ λp δN λp L2N (E ) ≥ min |∇u(x) · ω|N , [2.10] λ N AN ) and thus that (1 − δ)p |∇u(x) · ω|p dω dx, ( )! |u(x) − u(y)| λp L2N (x, y) ∈ N × N : ≥ λ R R N +1 p N N −1 |x − y| where the integral is over all points (x, ω) ∈ R × S with Z ∇u(x) · ω 6= 0, and by monotone convergence, ≤ C 0(p, N ) (M |∇u|)p (x)dx. N R p Z Z p 2N (1 − δ) p lim inf λ L (Eλ) ≥ |∇u(x) · ω| dω dx. For 1 < p < ∞, the maximal function theorem then implies λ→∞ N N N −1 R S " # u(x) − u(y) Since δ > 0 is arbitrary, we conclude that ≤ C (p, N )k∇uk p N . N L (R ) p +1 |x − y| M p ( N × N ) Z R R p 2N k(p, N ) p lim inf λ L (Eλ) ≥ |∇u(x)| dx λ→∞ N N The constant coming the maximal function theorem deteriorates R as p & 1. where k(p, N ) is deﬁned by [1.6]. 3. Proof of Theorem 1.2 It remains to establish that We now prove Theorem 1.2 and hence the lower bound in Z p 2N k(p, N ) p Theorem 1.1. lim sup λ L (Eλ) ≤ |∇u(x)| dx. [3.5] N N We will use the inequalities λ→∞ R

N From [3.2] we have |u(x) − u(y)| ≤ L|x − y| ∀x, y ∈ R [3.1] 2 L := k∇uk ∞ N |u(x) − u(y)| ≤ |∇u(x) · (x − y)| + A|x − y| with L (R ) and

2 N |u(x) − u(y) − ∇u(x) · (x − y)| ≤ A|x − y| ∀x, y ∈ R and thus if (x, y) ∈ Eλ we obtain [3.2] λr N /p ≤ |∇u(x) · ω| + Ar [3.6] 2 with A := k∇ uk ∞ N . L (R ) y−x N −1 N N −1 where again r = |y − x| and ω = ∈ . On the other Fix x ∈ R and a direction ω ∈ S . For a large positive num- |y−x| S N hand, if (x, y) ∈ E , we have from [3.1] that ber λ, consider the set Eλ(x, ω) consisting of all y ∈ R such λ that y − x is a positive multiple of ω and (x, y) ∈ Eλ. We will N /p determine two numbers R = R(x, ω, λ) and R = R(x, ω, λ) such λr ≤ L. [3.7] that Inserting [3.7] into [3.6] yields {x + rω : r ∈ (0, R]} ⊆ Eλ(x, ω) ⊆ {x + rω : r ∈ (0, R]}. L p/N Using polar coordinates, we then deduce that λr N /p ≤ |∇u(x) · ω| + A . [3.8] λ Z 1 N N N R(x, ω, λ) dω ≤ L {y ∈ R :(x, y) ∈ Eλ} In what follows we will consider only N N −1 S [3.3] 1 Z ≤ R(x, ω, λ)N dω. λ > L. [3.9] N N −1 S Observe that if dist(x, supp u) > 1, then From [3.2] we have N 2 1+ N {y ∈ R :(x, y) ∈ Eλ} = ∅. [3.10] |u(x) − u(y)| ≥ |∇u(x) · (x − y)| − A|x − y| ≥ λ|x − y| p Indeed by [3.7] and [3.9] we have, for any (x, y) ∈ E , provided λ that |x − y| ≤ 1. So if dist(x, supp u) > 1 and y ∈ N is such Ar + λr N /p ≤ |∇u(x) · ω| [3.4] R that (x, y) ∈ Eλ, then y ∈/ supp u, from which it follows that y−x N −1 N +1 where r := |y − x| and ω = |y−x| ∈ S . λ|x − y| p ≤ |u(x) − u(y)| = 0, i.e., x = y, which is a contra- Fix δ > 0 arbitrarily small. Then by [3.4], the conditions diction since (x, x) ∈/ Eλ.

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N N ≥ ≥ when C

n all and N 1 when and ) u ) |u |x p k u ) 1, ≤ 1, − ku |x ≤ (x 1 L 1−θ 1 when (x | − 1 − W θ ∈ C θ C hr xssaconstant a exists there hr xssaconstant a exists there − ) ( k p ) 1 R θ s https://doi.org/10.1073/pnas.2025254118 L θ − y (1, s = 1 ku 1 ku N 1 − 1 y ) ≤ p | u = ,p 1 p ≤ ) u | · N p p k ( u 1 1 k N ∞) θ p , p ∈ 1 1 = 1 p s (y R 1 +s ∞; L 1−/p p ( L 1−/p (y ≥ 1 +s N R < ∞ C = 1 ∞ p (1 + ) N ) ) 1 < 1 c with k∇u (

Teagmn sthe is argument (The ∞. ∞ ( ) p R

R < k∇u θ L Bei n Mironescu and [Brezis − ∀u N ∞ 1 N L (R p ) − p 1 (R ) (1 + θ k∇u ( k∇u s k N and ∈ R N θ 1 L 1−θ k Chn Dahmen, [Cohen, N ), p ) ×R 1 C L 1−θ p ×R 1 (R . 1 − c 1 k k ∞ ( ≥ N N R L 1/p = N L 1/p θ ) s N (R 1 ) 1 1 ) W 1 ) PNAS ( ( ∞ ) ˙ R p R . n o any for and C C N 0 N 1 N 1,1 ) ) < ≥ ) = = n the and , . where , | s Our 1. C C 1 [5.2] [5.1] [5.3] [5.7] [5.5] [5.6] [5.4] f6 of 5 < (N (N 1, ) )

MATHEMATICS where 0 < s < 1 and 1 < p < ∞ are deﬁned by [5.5]. ACKNOWLEDGMENTS. This work was completed during two visits of J.V.S. The proofs of Corollaries 4.1, 5.1, and 5.2 rely on our main to Rutgers University. J.V.S. thanks H.B. for the invitation and the Depart- Theorem 1.1 and the details will appear in a forthcoming ment of Mathematics for its hospitality. P.-L.Y. was partially supported by the General Research Fund CUHK14313716 from the Hong Kong Research article. Grant Council. H.B. is grateful to C. Sbordone who communicated to him the interesting paper by Greco and Schiattarella (listed below) which triggered Data Availability. There are no data underlying this work. our work. We are indebted to E. Tadmor for useful comments.

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