Families of functionals representing Sobolev norms

We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $\nu_{\gamma} (E_{\lambda,\gamma/p}[u])$ where \[ E_{\lambda,\gamma/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+\gamma/p}}>\lambda\Big\} \] and the measure $\nu_{\gamma}$ is given by $\mathrm{d} \nu_\gamma(x,y)=|x-y|^{\gamma-N} \mathrm{d} x \mathrm{d} y$. We provide characterizations which involve the $L^{p,\infty}$-quasi-norms $\sup_{\lambda>0} \lambda \, \nu_{\gamma} (E_{\lambda,\gamma/p}[u]) ^{1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $\lambda\to\infty$ when $\gamma>0$ and the limit for $\lambda\to 0^+$ when $\gamma<0$. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For $p>1$ the characterizations hold for all $\gamma \neq 0$. For $p=1$ the upper bounds for the $L^{1,\infty}$ quasi-norms fail in the range $\gamma\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $\gamma+1$. For $\gamma=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $\nu_0(E_{\lambda,0}[u])$.


Introduction
We are concerned with various ways in which we can recover the Sobolev seminorm ∥∇u∥ L p ‫ޒ(‬ N ) via positive nonconvex functionals involving differences u(x) − u(y).
We begin by mentioning two relevant results already in the literature.A theorem of H.-M. Nguyen [2006] (see also [Brezis and Nguyen 2018;2020]) states that, for 1 < p < ∞ and u in the inhomogeneous Sobolev space W 1, p ‫ޒ(‬ N ), and e is any unit vector in ‫ޒ‬ N .As shown in [Brezis and Nguyen 2018], (1-1) still holds for all u ∈ C 1 c ‫ޒ(‬ N ) when p = 1 but fails for general u ∈ W 1,1 ‫ޒ(‬ N ).The limit formula (1-1) may be compared to a theorem of [Brezis et al. 2021b], which states that, for all u ∈ C ∞ c ‫ޒ(‬ N ) and 1 ≤ p < ∞, one has where L 2N denotes the Lebesgue measure on ‫ޒ‬ N × ‫ޒ‬ N .Our first result, namely Theorem 1.1 below, provides an extension of (1-1) and (1-3) that unifies the two statements.Before we state the theorem, we introduce some notation that will be used throughout the paper.
First, for Lebesgue measurable subsets E of ‫ޒ‬ 2N = ‫ޒ‬ N × ‫ޒ‬ N and γ ∈ ‫,ޒ‬ we define (1-6) We will denote by Ẇ 1, p ‫ޒ(‬ N ), p ≥ 1, the homogeneous Sobolev space, i.e., the space of L 1 loc ‫ޒ(‬ N ) functions for which the distributional gradient ∇u belongs to L p ‫ޒ(‬ N ), with the seminorm ∥u∥ Ẇ 1, p := ∥∇u∥ L p ‫ޒ(‬ N ) .The inhomogeneous Sobolev space W 1, p is the subspace of Ẇ 1, p -functions u for which u ∈ L p , and we set ∥u∥ W 1, p := ∥u∥ L p + ∥∇u∥ L p .For p = 1 we will also consider the space ḂV(‫ޒ‬ N ) of functions of bounded variations, i.e., locally integrable functions u for which the gradient ∇u ∈ M belongs to the space M of ‫ޒ‬ N -valued bounded Borel measures and we put ∥u∥ ḂV := ∥∇u∥ M ; furthermore, let BV := ḂV ∩ L 1 .In the dual formulation, with C 1 c denoting the space of C 1 functions with compact support, For general background material on Sobolev spaces, see [Brezis 2011;Stein 1970].
In the proofs of these formulas one relates limits involving λν γ (E λ,γ / p [u]) 1/ p to (the absolute value of) limits of directional difference quotients δ −1 (u(x + δθ ) − u(x)) with increment δ = λ − p/γ , and in order to recover the directional derivative ⟨θ, ∇u(x)⟩ we need to let δ → 0, which suggests that we need to take λ → ∞ or λ ↘ 0 depending on the sign of γ .For the calculations see the proofs of Lemmas 3.2 and 3.3 below.
For a more detailed discussion we refer to Section 3F.See also Section 7B for a discussion about some related open problems.
Motivated by [Brezis et al. 2021b], we will also be interested in what happens to the larger quantity obtained by replacing the limits on the left-hand sides of (1-7) and (1-8) by sup λ>0 .This will be formulated in terms of the Marcinkiewicz space L p,∞ ‫ޒ(‬ 2N , ν γ ) (a.k.a.weak-type L p ) defined by the condition (1-10) As an immediate consequence of Theorem 1.1 we have, for -11) where C(N , p, γ ) is a positive constant depending only on N, p and γ .Moreover, the same conclusion holds for all u ∈ Ẇ 1, p ‫ޒ(‬ N ) when p > 1, with any γ ̸ = 0, and when p = 1, with any γ / ∈ [−1, 0].We shall show that the conditions in the last statement can in fact be relaxed; see the inequalities (1-14) and (1-16) below.In addition we have the important upper bounds for Q γ / p u, extending the case γ = N already dealt with in [Brezis et al. 2021b] The result in [Brezis et al. 2021b] states that, for every N ≥ 1, there exists a constant C(N ) such that ) and all 1 ≤ p < ∞.In light of Theorem 1.1, it is natural to ask whether one can replace the limits on the left-hand sides of (1-7) and (1-8) by sup λ>0 and still obtain a quantity that is comparable to ∥∇u∥ p L p ‫ޒ(‬ N ) .As suggested by Theorem 1.1 the answer to our question is sensitive to the values of γ and p. Theorem 1.3.Suppose that N ≥ 1, 1 < p < ∞ and γ ∈ ‫.ޒ‬ Then the following hold: ) and we have the inequality (
Historical comments.Some special cases of the above quantitative estimates have been known.Estimate (1-13) for γ = − p and 1 < p < ∞ was discovered independently by H.-M. Nguyen [2006], and by A. Ponce and J. Van Schaftingen (unpublished communication to H. Brezis and H.-M. Nguyen), both relying on the Hardy-Littlewood maximal inequality. A. Poliakovsky [2022] recently proved generalizations of results in [Brezis et al. 2021b] to Sobolev spaces on domains; moreover, he obtained Theorems 1.3 and 1.4 in the special case γ = N under the additional assumption that u ∈ L p .Other far-reaching generalizations to one-parameter families of operators were obtained by Ó. Domínguez and M. Milman [2022].
The case γ = 0. We shall now return to the necessity of the assumption γ / ∈ [−1, 0] in parts of Theorems 1.1, 1.3 and 1.4.When γ = 0, the bounds for [Q γ / p u] L p,∞ ‫ޒ(‬ 2N ,ν γ ) fail in a striking way.We begin by formulating a result illustrating this failure, which also gives a characterization of the seminorm in the Lipschitz space Ẇ 1,∞ .
In view of other known results [Brezis 2002;Brezis et al. 2021a] on how to recognize constant functions, a natural question arises whether the hypothesis on the local integrability of ∇u in the corollary could be relaxed; one can ask whether the constancy conclusion holds for all locally integrable functions satisfying ν 0 (E λ,0 [u]) < ∞ for all λ > 0. However, the following example shows that such an extension fails (for details, see Lemma 5.2).
More on counterexamples.We now make more explicit the exclusion of the parameters γ ∈ [−1, 0) in part (c) of Theorem 1.1 and in (1-15).We shall show in Section 6B that for γ ∈ (−1, 0) these negative results can be related to self-similar Cantor subsets of ‫,ޒ‬ of dimension 1 + γ .
The case N = 1 = −γ plays a special role and is excluded in the strongest statement (iii) since for all compactly supported u ∈ Ẇ 1,1 ‫)ޒ(‬ one has ν −1 (E λ,−1 [u]) < ∞ for all λ > 0 (see Lemma 6.5 below).The proofs of existence of counterexamples are constructive and the Baire category statements will be obtained as rather straightforward consequences of the constructions.
Outline of the paper.In Section 2 we provide the upper bounds for [Q γ / p u] L p,∞ ‫ޒ(‬ 2N ,ν γ ) , i.e., the proof of inequalities (1-13) and (1-15) in Theorems 1.3 and 1.4.We first derive these for a dense subclass, relying on covering lemmas, and then extend in Sections 2C and 2D to general Ẇ 1, p and ḂV-functions.
In Section 3 we derive the limit formulas of Theorem 1.1; specifically in Section 3B we prove the sharp lower bounds involving a lim inf λ p ν γ (E λ,γ / p [u]) for general functions in Ẇ 1, p and in Section 3C we obtain the sharp upper bounds for lim sup λ p ν γ (E λ,γ / p [u]), under the assumption that u ∈ C 1 is compactly supported.Then in Section 3D we extend these limits to general Ẇ 1, p functions.In Section 3F we show that the limit formulas for Ẇ 1,1 do not extend to general ḂV functions and prove Proposition 1.2.In Section 4 we prove the reverse inequalities (1-14) and (1-16) in Theorems 1.3 and 1.4.In Section 5 we prove Theorem 1.5 on a characterization of the Lipschitz norm and also discuss Example 1.7.In Section 6 we provide various constructions of counterexamples and in particular prove Theorem 1.8.We discuss some further perspectives and open problems in Section 7.
2A.The bound (1-13) via the Hardy-Littlewood maximal operator.Following [Brezis et al. 2021b], one can prove the result of Theorem 1.3 for p > 1 by an elementary argument involving the Hardy-Littlewood maximal function M|∇u| of |∇u|; however, the behavior of the constants as p ↘ 1 will only be sharp in the range −1 ≤ γ < 0.
Proposition 2.1.Let N ≥ 1 and 1 < p < ∞.There exists a constant C N such that, for all γ ̸ = 0 and all u ∈ Ẇ 1, p ‫ޒ(‬ N ), (2-1) Proof.We assume first that u ∈ C 1 and that ∇u is compactly supported.As in [Brezis et al. 2021b, Remark 2.3], one uses the Lusin-Lipschitz inequality and observes that (2-2) implies As a consequence Direct computation of the inner integral (distinguishing the cases γ > 0 and γ < 0) yields Inequality (2-1) follows then from the standard maximal inequality ∥M f ∥ ).The extension to general Ẇ 1, p functions will be taken up in Section 2C.□ 2B.The case γ ∈ ‫ޒ‬ \ [−1, 0].We shall prove the following more precise versions of the estimates (1-13) and (1-15) when γ / ∈ [−1, 0], with constants that stay bounded as p ↘ 1; indeed we cover all p ∈ [1, ∞).We denote by σ N −1 the surface area of the sphere ‫ޓ‬ N −1 .In the proof of the following theorem we will first establish the estimates for functions u ∈ C 1 ‫ޒ(‬ N ) whose gradient is compactly supported.The extension to Ẇ 1, p and ḂV will be taken up in Sections 2C and 2D.
The proof of Theorem 2.2 relies on the following proposition, in which [x, y] ⊂ ‫ޒ‬ N denotes the closed line segment connecting two points x, y ∈ ‫ޒ‬ N .
There exists an absolute constant C > 0 such that, for every N ≥ 1 and every f ∈ C c ‫ޒ(‬ N ): (2-7) Indeed, to deduce Theorem 2.2 from Proposition 2.3 one argues as in the proof of (1-12) in [Brezis et al. 2021b]; for u ∈ C 1 ‫ޒ(‬ N ) and 1 ≤ p < ∞, one has |∇u| p ds |x − y| p−1 for all x, y ∈ ‫ޒ‬ N , which implies Hence for u ∈ C 1 ‫ޒ(‬ N ) whose gradient is compactly supported, one establishes Theorem 2.2 by applying Proposition 2.3 with f := λ − p |∇u| p .The extension to u ∈ Ẇ 1, p will be taken up in Section 2C.
Proof of Proposition 2.3.As in the proof of [Brezis et al. 2021b, Proposition 2.2], using the method of rotation, we only need to prove Proposition 2.3 for N = 1.Indeed, where for every ω ∈ ‫ޓ‬ N −1 and every x ′ ∈ ω ⊥ , f ω,x ′ is a function of one real variable defined by The innermost double integral can be estimated by the case N = 1 of Proposition 2.3, and Thus from now on, we assume On the other hand, suppose now γ < −1.Without loss of generality, assume f ≥ 0 on ‫.ޒ‬In addition, we may assume that f is not identically zero, for otherwise there is nothing to prove. Let Then by symmetry, and it suffices to estimate the latter integral.
In what follows we will need to always keep in mind that in view of our assumption γ < −1 we have We will now use a simple stopping-time argument based on the fact that for all c ∈ ‫ޒ‬ the continuous function We construct a finite sequence of intervals I 1 , . . ., I K , that are disjoint up to endpoints, that cover supp f = [a, b], and that satisfy Indeed, we may take a 1 := a, and a 2 > a 1 to be the unique number for which and set If a 2 < b, we may now repeat, and take I 2 := [a 2 , a 3 ], where a 3 > a 2 is the unique number for which (a 3 − a 2 ) −(γ +1) a 3 a 2 f = 1 2 .Note that the a i 's chosen as such satisfy +1) .This shows that in finitely many steps, we would reach a K +1 ≥ b for some K ≥ 1, with a K < b if 1 ≤ K .Then we have our sequence of disjoint (up to endpoints) intervals I 1 , . . ., and satisfy (2-8).We also write I 0 := (−∞, a 1 ] and This being trivially the case when i ∈ {0, K + 1}, we consider the case i ∈ {1, . . ., K }: any x, y ∈ I i satisfy It follows thus that (again we used γ < −1 so that −(γ +1) > 0 here), from which it follows that (x, y) ̸ ∈ E + ( f, γ ).Combining this with a similar argument for i ∈ {1, K + 1}, we get that if (x, y) (The computation of these integrals uses our assumption γ + 1 < 0.) Summing the estimates, we get in view of (2-9) We have thus completed the proof of (2-7) under the assumption γ < −1 and N = 1.□ 2C.Proof of Proposition 2.1 and Theorem 2.2 for general Ẇ 1, p functions.We use a limiting argument, together with the following fact: if u ∈ Ẇ 1, p ‫ޒ(‬ N ), N ≥ 1, and 1 ≤ p < ∞, then there exists a Lebesgue measurable set X ⊂ ‫ޒ‬ 2N , with L 2N (X ) = 0, so that, for every (x, h) ∈ ‫ޒ‬ 2N \ X , we have (2-10) Indeed, both sides are measurable functions of (x, h) ∈ ‫ޒ‬ 2N , and if X is the set of all (x, h) where the two sides are not equal, then X is a measurable subset of ‫ޒ‬ 2N , and the assertion will follow from Fubini's theorem if, for every fixed h ∈ ‫ޒ‬ N , we have ) such that ∇u n are compactly supported, and (2-11) Indeed if N > 1 and p ≥ 1, or if N = 1 and p > 1, then this follows from the density of C ∞ c ‫ޒ(‬ N ) in Ẇ 1, p ‫ޒ(‬ N ) as asserted in [Hajłasz and Kałamajska 1995] (in this case one may choose [Hajłasz and Kałamajska 1995]); the issue is that if ∇u is supported in a convex set in ‫ޒ‬ N , N ≥ 2, then u is constant in the complement of the set, but this fails for N = 1 since the complement of a bounded interval has two connected components.On the other hand, in the anomalous case N = 1 and p = 1, one can choose an approximation of the identity to get a sequence v n of C ∞ c functions on ‫ޒ‬ such that ∥v n − u ′ ∥ L 1 ‫)ޒ(‬ → 0. One can then take u n (x) := x 0 v n (t) dt, and (2-11) follows with u ′ n = v n being compactly supported (even though u n may not be compactly supported).
Let, for R > 1, By monotone convergence it suffices to prove Under the assumptions of Proposition 2.1 and Theorem 2.2 on p and γ , since Moreover, the sequence Q γ / p u n converges to Q γ / p u in L p (K R ) as n → ∞.Indeed, using (2-10) we may write for L 2N a.e.(x, y) ∈ ‫ޒ‬ 2N , and similarly for u n in place of u, which allows us to estimate By passing to a subsequence if necessary, we may assume that which implies 2D. Proof of Theorem 2.2 for ḂV-functions.We choose a sequence by dominated convergence.Also here we used ∥ρ n * ⃗ φ∥ ∞ ≤ ∥ ⃗ φ∥ ∞ for the last inequality.Combining these two limiting identities with Theorem 2.2 we get the desired inequalities with E λ,γ 3A.A Lebesgue differentiation lemma.Our argument uses the following standard variant of the Lebesgue differentiation theorem.For lack of a proper reference, a proof is provided for the convenience of the reader.
Lemma 3.1.Let u ∈ Ẇ 1,1 ‫ޒ(‬ N ) and let {δ n } be a sequence of positive numbers with lim n→∞ δ n = 0. Then Proof.If u ∈ C 1 with compact support the limit relation clearly holds for all (x, h).We shall below consider for each θ ∈ ‫ޓ‬ N −1 the maximal function which is well-defined for all θ , a measurable function on ‫ޒ‬ N × ‫ޓ‬ N −1 , and satisfies a weak-type (1, 1) inequality From (2-10) we get that, for every n ≥ 1, × A M such that the identity holds for all (x, h) ∈ N ∁ and all n ≥ 1.It suffices to show that, for every α > 0, ε > 0, Since the asserted limiting relation holds for v, we see that the expression on the left-hand side of (3-1) is dominated by The lower bounds for lim inf λ p ν γ (E λ,γ / p [u]).We use Lemma 3.1 to establish lower bounds, relying on an idea in [Brezis and Nguyen 2018], where the case γ = −1 was considered.
Then the following hold: (iii) The statement in part (i) continues to hold for u ∈ C 1 ‫ޒ(‬ N ) whose gradient is compactly supported.
Remark 3.4.The subtlety in part (iii) above is only relevant in dimension N = 1, since if N ≥ 2, then any function in C 1 ‫ޒ(‬ N ) with a compactly supported gradient is constant outside a compact set.
The case γ > 0. We assume that ∇u is compactly supported.To prove part (iii) (and thus part (i)) assume . (3-4) , then writing y = x + r ω with r > 0 and ω ∈ ‫ޓ‬ N −1 , we have since ∇u is uniformly continuous on ‫ޒ‬ N , we have ρ(r ) ↘ 0 as r ↘ 0. This, together with the first implication of (3-5), shows Let B be a ball centered at the origin containing supp(∇u), and let B be the expanded ball with radius 1 + rad(B).Then for x / ∈ B, we have Q γ / p u(x, y) = 0 for every y with |x − y| ≤ 1, and (3-5) shows Then by (3-7), Letting λ → ∞ we get lim sup and hence the assertion.

3E. Conclusion of the proof of Theorem 1.1.
In Section 3D we proved parts (a) and (b) of Theorem 1.1.The lower bound for the lim inf in part (c) has been established in Lemma 3.2(ii), and the limiting equality for u ∈ C 1 c ‫ޒ(‬ N ) when p = 1 and −1 ≤ γ < 0 follows by combining that with the upper bound for the lim sup in part (ii) of Lemma 3.3.The proof of the negative result in part (c) of the theorem (generic failure for p = 1, −1 ≤ γ < 0) will be given in Proposition 6.6 below.□ 3F.On limit formulas for ḂV(‫-)ޒ‬functions: the proof of Proposition 1.2.When p = 1, Poliakovsky [2022] asked whether (1-7) still holds for u ∈ ḂV(‫ޒ‬ N ) instead of Ẇ 1,1 ‫ޒ(‬ N ) if γ = N.More generally, one may wonder whether it is possible that, for all u ∈ ḂV(‫ޒ‬ N ), one has We show that this is not the case.First, when −1 ≤ γ < 0, Theorem 1.8(i) (proved in Proposition 6.3 below) shows that even if u ∈ Ẇ 1,1 ‫ޒ(‬ N ), it may happen that lim λ→0 + λν γ (E λ,γ [u]) = ∞.So (3-14) cannot hold for all u ∈ ḂV(‫ޒ‬ N ) for such γ .
Proof.First consider the case which follows from a change of variables s = x − y, t = x + y: when γ > −1, one has A similar calculation shows that if u = 1 I is a characteristic function of a bounded open interval (so that we also have Now consider the case N ≥ 2. Let be a bounded convex domain in ‫ޒ‬ N with smooth boundary and u = 1 .Then u ∈ ḂV(‫ޒ‬ N ) with ∥∇u∥ M = L N −1 (∂ ).The method of rotation shows ∈ ω ⊥ , since is convex and every line only meets ∂ at at most two points.Thus (3-16), (3-18) and the dominated convergence theorem allow one to show that and using (3-17) in place of (3-16) we obtain the same conclusion with lim λ→∞ replaced by lim λ→0 + if γ < −1.It remains to observe that This holds by Fubini's theorem if u = 1 is replaced by u ε := u * ρ ε , where ρ ε is a suitable family of mollifiers, because the left-hand side is then just But for every ω ∈ ‫ޓ‬ N −1 , and almost every x ′ ∈ ω ⊥ (as long as t → x ′ + tω parametrizes a line L ω,x ′ that is either disjoint from , or intersects ∂ transversely at two different points), we have lim The validity of (3-21) is clear if L ω,x ′ does not intersect , while if L ω,x ′ intersects ∂ transversely at two different points, then we can choose a coordinate system so that ω = (0, . . ., 0, 1), and assume that for some open neighborhood U of x ′ in ω ⊥ , the intersection of U × L ω,x ′ with takes the form for some smooth functions φ 1 and φ 2 of y ′ ∈ U.Then, for ε > 0 sufficiently small, ޒ‬ .This proves (3-21), and then the dominated convergence theorem allows one to conclude the proof of (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20).□ Remark.The identity (3-19) for u = 1 can be derived from Crofton's formula for rather general (not necessarily convex) domains .See [Federer 1969, Chapter 3.2.26],which showed that when ∂ is rectifiable, then its (N −1)-dimensional Hausdorff measure , where Federer 1969, Chapter 2.10.15] as is the number of points x ∈ ∂ so that px = y, and according to [Federer 1969, Chapter 3.2.13].It follows that, for u = 1 ,

Using the definition of weak derivative we see by a limiting argument that the conclusion sup
In order to establish our estimate for bounded functions we will use Lorentz duality in the following form: if F, G are measurable functions on ‫ޒ‬ 2N , then, for any 1 < q < ∞, we have where 1/q + 1/q ′ = 1, here F * (t) := inf{s > 0 : ν γ ({|F| > λ}) ≤ s} is the nonincreasing rearrangement of F, and similarly for G * (t); see [Hunt 1966;Stein and Weiss 1971].Indeed, (4-2) follows by noticing that . First we consider the case γ > 0. For sufficiently small s > 0, define Furthermore, since γ > 0, we have Recall θ = s/(1 + γ / p).Thus as s → 0 + , we have lim sup Since this upper bound holds uniformly over all R > 0, this concludes the argument for the case γ > 0.
For the more substantial part assume λ < ∥∇u∥ ∞ , where ∥∇u∥ ∞ may be finite or infinite.We need to show that ν We now consider the set S ε of all spherical balls S ⊂ ‫ޓ‬ N −1 with positive radius and the property that ⟨θ 1 , θ 2 ⟩ > 1 − ε for all θ 1 , θ 2 ∈ S. By pigeonholing there exists a spherical ball S ∈ S ε and a Lebesgue measurable subset F ⊂ F 0 such that L N (F) > 0 and ∇u(x)/|∇u(x)| ∈ S for all x ∈ F. For the remainder of the argument we fix this spherical ball S; we denote by σ (S) its spherical measure.
Finally consider the case γ = −1.We now choose v m as in (6-3) and -22) In analogy to (6-20) we now use (using (6-22) in the last inequality).This, together with our assumption on m(n), implies that when n → ∞, as desired.□ The next proposition is relevant for part (ii) of Theorem 1.8.Proposition 6.4.Suppose −1 ≤ γ < 0. Then there exists a compactly supported u If in addition N ≥ 2 or −1 < γ < 0 there exists u with the above properties and Proof.Consider first the case −1 < γ < 0. We choose for n ∈ ‫ގ‬ and with these choices of R n and m(n) and f m as in (6-17) and (6-11) we define again −2 for small x.Also, because of the choices of R n , we see that u is smooth away from 0.
Fix λ > 0. Since lim n→∞ R N +γ n n 2 = 0, we may choose n 0 such that Hence by the same rescaling argument as in (6-21), we obtain If n ≥ n 0 then this gives For the case γ = −1 and N ≥ 2, define u as in (6-23) but with the choice of the parameters R n , m(n) as in (6-27) to obtain a compactly supported u ∈ W 1,1 satisfying (6-24).We now fix λ > 0 and note that when N ≥ 2 we have λR Finally, follows from (6-26), and the latter was proved if −1 < γ < 0 or N ≥ 2. It remains to consider the case N = 1, γ = −1.We define u as in the previous paragraph.The above calculation shows that ν −1 (E λ,−1 [u]) ≥ cm(n) provided that λ < 1/n 2 which establishes (6-25) in this last case.□ The case N = 1, γ = −1 plays a special role.The following lemma shows that the conclusion (6-26) in Proposition 6.4 fails in this case.
For the proof of Proposition 6.6 we use an elementary estimate for the intersections E λ,γ [u] ∩ ℓ .
To show that the closed set V(m, j) is nowhere dense when −1 ≤ γ < 0, we need to verify that for every u ∈ V(m, j) and ε 1 > 0 there exists f ∈ W 1,1 ‫ޒ(‬ N ) such that ∥ f − u∥ W 1,1 ‫ޒ(‬ N ) < ε 1 and f / ∈ V(m, j).To see this we use Proposition 6.4, according to which there exists a compactly supported W 1,1 function f 0 for which ν γ (E λ,γ [ f 0 ]) = ∞ for all λ > 0. It is then clear that for every λ > 2 j , for all j ∈ ‫.ޚ‬The proposition is proved.□ To include a result of generic failure of the limiting relation in the case N = 1, γ = −1 we give Proposition 6.8.Let −1 ≤ γ < 0. Let Then W is of first category in W 1,1 , in the sense of Baire.
7B.Other limit functionals.Our results, combined with the various developments presented in [Brezis and Nguyen 2018;2020;Nguyen 2007;2011], suggest several possible directions of research.