A note on zero sets of fractional sobolev functions with negative power of integrability

We extend a Poincar\'{e}-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincar\'{e}-style inequality.


Introduction
Let Here and henceforth, for a measurable set A ⊂ R n we denote the mean value integral In [7] Jiang and Lin showed that if f ∈ W 1,p (Ω), then H s (Σ) = 0 where s = max{0, n − pα p+α }. They were motivated by the analysis of rupture sets of thin films, which is described by a singular elliptic equation. We do not go into the details of this and instead, for applications we refer to, e.g., [3,6,2,8].
Here, we use the following definitions for the (fractional) Sobolev space. For more on these we refer to, e.g., [4,1,10].
The homogeneous W σ,p -norms are defined as follows: For σ ∈ (0, 1) we define the Slobodeckij-norm, To prove Theorem 1.1, the case p ≤ n/σ is the relevant one, since for the other cases we can use the embedding into the Hölder spaces, see [7]. We have the following extension to fractional Sobolev spaces of a Poincaré-type inequality from [7]. Theorem 1.3. For any θ > 0, σ ∈ (0, 1], p ∈ (1, n/σ], s ∈ (n − σp, n], there is a constant C > 0 such that the following holds for any R > 0: Let B R be any ball in R n with radius R, f ∈ W σ,p (B R ) and assume that there is a closed set T ⊂ B R such that and for any ball B r with some radius r > 0, Then, In [7] this was proven for the classical Sobolev space W 1,p , using an argument based on the p-Laplace equation with measures and the Wolff potential.
Our argument, on the other hand, is completely elementary and adapts the classical blow-up proof of the Poincaré inequality, see Section 2.
Once Theorem 1.3 is established, one can follow the arguments in [7] to obtain Theorem 1.1. These rely heavily on the theory of Sousslin sets, [9], to find the closed set T ⊂ Σ with the condition (1.2) and (1.3) satisfied. Those arguments are by no means elementary, but we were unable to remove them in order to show that H s (Σ) = 0. However, if one is satisfied in showing that H t (Σ) = 0 for any t > s, then there is a completely elementary argument, the details of which we will present in Section 3. There, we prove the following "pointwise" Poincaré-style inequality, from which the suboptimal Hausdorff dimension estimate easily follows, see Corollary 3.1.
, there exists C > 0, such that the following holds. Let f ∈ L p loc , and assume x ∈ R n , such that Acknowledgments. The author thanks P. Haj lasz for introducing him to Jiang and Lin's paper [7].

Poincaré Inequality: Proof of Theorem 1.3
By a scaling argument, Theorem 1.3 follows from the following there is a constant C > 0 such that the following holds: as well as H s (T ∩ B r ) ≤ θr s for any ball B r with radius r > 0.
Proof. We proceed by the usual blow-up proof of the Poincaré inequality: Assume the claim is false, and that for fixed θ, p, s, σ for any Replacing f k by f k f k p (note that this does not change the definition and size of T k ), we can assume w.l.o.g.
In particular, f k is uniformly bounded in W σ,p , and by the Rellich-Kondrachov theorem, up to taking a subsequence, f k converges strongly in L p , and weakly in W σ,p to some and setting g k := |B 1 | 1 p f k , we have found a sequence such that and H s (T k ∩ B r ) ≤ θr s for any ball B r . This is a contradiction to Lemma 2.2.
We used the following lemma, which essentially quantifies the intuition, that a function approximating 1 in W σ,p cannot be zero on a large set. By extension, we also can assume that f k − 1 → 0 in W σ,p (R n ), and f k ≡ 1 on R n \B 2 .
On the one hand, we have On the other hand, up to picking a subsequence, we can assume the existence of R k ∈ (0, 1), for k ∈ N, and lim k→∞ R k = 0, such that Since for any point x ∈ T k we have that lim t→0 -Br f k (x) = 0, we expect the the average (fractional) gradient around x to be fairly large. More precisely, we have the following Claim. There is a uniform constant c s,σ,p > 0, such that the following holds: For any x ∈ T k , there exists ρ = ρ k,x ∈ (0, R k ) such that Of course, we only have to show the first inequality, the second inequality is the classical Poincaré inequality.
For the proof let us write f instead of f k . Then, since for x ∈ T , Consequently, for any ε > 0, there has to be some c ε > 0 and some l ∈ N such that because if the opposite inequality was true for all l ∈ N we would have which is false for c ε small enough.
For any k we cover T k by the family Since T ⊂ B 2 is closed and bounded, i.e. compact, we can find a finite subfamily still covering all of T k , and then using Vitali's (finite) covering theorem, we find a subfamilyF k ⊂ F k of disjoint balls B ρ (x), so that the union of the B 5ρ covers all of T k . We use thisF k as a cover for an estimate of the Hausdorff measure: 3. An elementary proof for the suboptimal case We start with the proof of the pointwise inequality, Lemma 1.4.
Proof. First, let us show the claim for p = 1: Fix R, ε > 0, f ∈ L 1 loc and assume x = 0. W.l.o.g., f ≥ 0. Set and C ε := R −ε τ −1 . Assume by contradiction that the claim was false, i.e. assume that for any ρ ∈ (0, R), Then for any K ∈ N, Setting now for l ∈ Z, a l := - the above equation applied to ρ = 2 l R reads as In particular for any L ∈ N, Under the additional assumption that In particular, for any i ∈ N, Since c i is monotonically increasing, This proves Lemma 1.4 for p = 1.
Let now V R := {B ρ (x) : x ∈ Σ, ρ < R, (3.6) holds}. Any countable disjoint subclass U R ⊂ V R satisfies By the Besicovitch covering theorem, as in, e.g., [5,Theorem 18.1], we find for any R a countable subclass U R ⊂ V R , such that any point of Σ is covered at least once, and at most a fixed number of times. Thus,