Bourgain-Brezis-Mironescu formula for $W^{s,p}_q$-spaces in arbitrary domains

Under certain restrictions on $s,p,q$, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces $W^{s,p}_q$. In this article, we show that the Bourgain-Brezis-Mironescu formula holds for $W^{s,p}_q$-seminorms in arbitrary domain. This addresses an open question raised by Brazke-Schikorra-Yung in [Bourgain-Brezis-Mironescu convergence via Triebel-Lizorkin spaces; Calc. Var. Partial Differential Equations; 2023].

Our first interest lies in works regarding the domain.That is, how far the smooth-bounded condition, on the domain, can be relaxed.In this direction, we mention three works.The first is due to Lioni-Spector [43,44].They showed that for an arbitrary domain Ω, and f ∈ L p (Ω), lim where ( 2) The second result is due to the author with Bal-Roy [5], where it has been shown that the BBM-formula [8], holds if we take Ω to be a W 1,p -extension domain.The third result is due to Drelichman-Duran [27].They showed that for 1 < p < ∞, and an arbitrary bounded domain Ω, and any τ ∈ (0, 1), we have The second direction of work regarding the BBM-formula, that we are interested in, is its extension for Triebel-Lizorkin spaces.The first work in this direction was done by Brazke-Schikorra-Yung [12].They explained via examining thoroughly various constants of embeddings that although F s p,p = W s,p , when s ∈ (0, 1) and F s p,2 = W 1,p it makes sense for the scaled W s,p seminorm to converge to W 1,p -seminorm, even when p = 2.They posed the open problem (see [12,Question 1.12]) about the asymptotic constant in the identification of the • W s,p q ≈ • F s p,q .The current article addresses this question by showing the asymptotic behaviour (as s → 1−) of W s,p q -seminorms.Similar studies has been done, when 1 < q < p < ∞, in [22] for R N , and in [62, Theorem 6.1] for a special class of bounded extension domains (called (ε, ∞)-domains).
We concentrate our focus on the following seminorm (for some τ ∈ (0, 1)) , as one can then extend the above question for arbitrary bounded domains, motivated by [27].We go one step further and show that the boundedness of the domain is not necessary.Our main results are the following: Then there is a constant K = K(N, p, q) > 0 such that for any f ∈ W 1,p (Ω), we have, Remark 3. Before proceeding further, let us discuss some difficulties that arise here, and strategies for overcoming them.The proof of the main results roughly follow the outline of [8].However, there are certain obstacles to that path.The first obstacle arises when we want to apply dominated convergence theorem to interchange limit and integral.Similar difficulty was faced and overcome in [27], but we had to take a different route (see lemma 9) for this purpose.The introduction of the second exponent q forces us to deviate from the usual route again; The case q ≤ p is rather easy to handle, for the case p < q, a careful use of Sobolev embedding is needed.To take into account the case where the domain Ω is unbounded, we need to restrict the seminorm further and define some new fractional Sobolev spaces (see eq. ( 5)) and prove a version of the main result theorem 1 in that context (see theorem 14), and then finally derive the proof of the main results from there.
The article is organised as follows: In section 2, we list some preliminary results, already known in literature, which shall be useful for the proof of our main results.In section 3, we introduce a variant of fractional Sobolev spaces and prove some relevant embedding results.In section 4 we prove the main results in the context of these new spaces (see theorems 14 and 15).Finally in section 5, we prove theorems 1 and 2.

Preliminary Results
For the sake of completeness, we first state the well-known Sobolev inequality (also known as (q, p)-Poincaré inequality): 1), and one of the following hold (1) p < N , and q ≤ N p N −p , (2) p = N , and q < ∞ (3) p > N .Then there is a constants C = C(p, q, N ) > 0 such that the following holds for any f ∈ W 1,p (B(0, t τ )): The following lemma was established in [20] for 1 < p < ∞; the p ≥ 1 case can be found in [58, Chapter-VI, Theorems 5 and 5'].
The following result can be found in Proposition 9.3 and Remark 6 of [13].
Next we list a special case of Proposition 2/(ii) of [59, Chapter 2.3.3],combined with the fact that The following result is taken from Lemma 8 of [5].
Lemma 8. Let Ω ⊆ R N be open and λ > 0 be sufficiently small.Then there is a bounded open set Ω * λ with smooth boundary such that The next result can be found in Theorem 2.1 of [40].It will play a crucial in this article. Then

Fractional Sobolev space with restricted internal distance
Fix R > 0 and τ ∈ (0, 1) once and for all.Denote δ x,R,τ = min{R, τ dist(x, ∂Ω)}.We shall often drop the R and τ in the above notation and write δ x to denote δ x,R,τ .
Remark 10.If the function x → dist(x, ∂Ω) is bounded in Ω, we can choose R > 0 large enough, so that δ x = τ dist(x, ∂Ω).Then the particular case p = q of theorems 14 and 15 are similar to the results proved in [27], but here Ω need not be a bounded domain; for example, it can be a cylindrical domain or any open subset of We shall need some embedding results for these new fractional Sobolev spaces for our purpose.As expected, the case q ≤ p and p < q are treated separately.
Proof.We have The last inequality follows from the absolute continuity on lines of the W 1,p -functions.We now have, after a change of variable y = x + th, (using that B(x, tδ x ) ⊂ B(x, δ x )) Note that in the above inequality, ∇f is required to be defined only inside D. So, we shall take a 0-extension of ∇f outside D. Since we have p q ≥ 1, we can use Young's convolution inequality to get (sq − q + 1) (sq − q + 1) According to either p < N or p ≥ N , fix β ∈ (0, 1), depending on p, q, N , such that (q, βp)-type Poincaré inequality (lemma 4) is satisfied.We use this to estimate I 1 below.First, we change the order of integration between t and h, then apply lemma 4.
As in the proof of the previous lemma, we shall take a 0-extension of ∇f outside D. Now using Hardy-Littelewood maximal inequality, we get Combining eqs.( 9) and ( 10), we get ( 11) Again, we can estimate I 2 , in similar way as above with δ x in place of t.We have a better estimate this time.Also, we can apply (q, p)-Poincaré inequality this time.
4. BBM formula for Ŵ s,p q -seminorms First, we state the following result whose proof can be found in the proof of Theorem of [8] as the quantity δ x is bounded by R.

Lemma 13.
Let Ω ⊂ R N be any open set, 1 ≤ q < ∞, 0 < s < 1.Then for any f ∈ C 2 (Ω), we have for all x ∈ Ω, (13) lim We now prove the following BBM-type results which are closely related with theorems 1 and 2.
Step-1: We show that it is enough to prove eq. ( 14) Since we have assumed that eq. ( 14) holds for functions in C 2 (Ω) ∩ W 1,p (Ω), for s > 1 2 , we have ( 16) Using triangle inequality, and then eq. ( 16) followed by eq. ( 15) and either lemma 11 or lemma 12, we get The proof of step-1 follows. Step-2: In view of the previous step, it is now enough to assume that f ∈ C 2 (Ω) ∩ W 1,p (Ω) and prove eq. ( 14).Let us take an arbitrary sequence s n ∈ (0, 1) such that s n → 1− as n → ∞.Set , and F (x) := K|∇f (x)| p .Also note that, lemma 13 implies that F n → F pointwise a.e in Ω.To complete the proof, it is enough to show that We shall apply lemma 9 on F n to show that the interchange of limit and integral is valid.
For any i ∈ N, consider the sets We need to verify the the hypotheses of lemma 9. First, note that for x ∈ Ω i , h ∈ B(0, δ x ), t ∈ (0, 1), we have . Thus, we have using triangle inequality and then mean value inequality, Since f is continuous in the closure of the bounded open set Ω i 2 , we have the hypothesis (1) of lemma 9 satisfied for sufficiently large i ∈ N.
Note that, to show that hypothesis (2) of lemma 9 is satisfied, it is enough to show that lim i→∞ lim n→∞ Ω\Ω2i F n (x)dx = 0.
We start with an arbitrary x ∈ Ω \ Ω 2i , h ∈ B(0, δ x ) and t ∈ (0, 1).There can be two cases: . Moreover, we can assume R < i without loss of generality.We have |x Hence we always have (17) x From eq. ( 17), we get

Now we apply lemma 11 or lemma 12 with
Hence we can integrate eq. ( 13) and interchange the limit and the integral to get the result.
Proof of Theorem 15.We divide the proof into two parts.First, we prove it for a particular case with a bit stronger assumptions, and then give the general proof.
Step-1: Ω is bounded with Ω ⊆ B(0, λ), and Lp,q (f ) := lim Extend f by 0, outside Ω.From the proof of Theorem 2 and 3 in [8] we can see that for any i = 1, 2, • • • , N , and where, We estimate J 1,s using Fubuni's theorem to change the order of integration, then using Hölder's inequality twice, first with respect to the measure dx |x−y| N +sq−q and then with respect to dy.We get Now using the hypothesis of Step-1, we have Using Hölder's inequality, we estimate J 2,s as in [8] to get Using eqs.(18) to (20), we get Hence by lemma 6, the result follows.
Step-2: We now prove the theorem in full generality.For 1 < p < ∞, define X 1,p (Ω) := W 1,p (Ω), and X We also have, for s > 1 2 , and R > 1 n , frfom the hypotheses, and Ω n are bounded domains, we have From Step-1, we can conclude that f ∈ X 1,p (Ω n ) for all n.Further the X 1,p -seminorms are uniformly bounded (independent of n) as can be seen from the following calculation, where we use theorem 14, The proof follows from the observation that K .

Proof of Theorems 1 and 2
Note that theorem 2 is a straightforward consequence of theorem 15, as L p,q (f ) ≤ L * p,q (f ).Theorem 1 is also a consequence of theorem 14, but it requires a bit more work.To complete the proof of theorem 1, we only need the following lemma: Ω) and R > 0. Assume one of the following conditions (1) 1 ≤ q ≤ N p N −p with p < N , (2) 1 ≤ q < ∞ with p ≥ N .Then eq. ( 14) implies eq.(4).
In order to prove this, we first prove a bit more general result.
Proof of Proposition 17.Note that, since dx, from eq. ( 14) we have lim We focus on the reverse inequality.Observe that for x, y ∈ Ω, δ x < |x − y| ≤ τ dist(x, ∂Ω) implies R < |x − y| ≤ τ dist(x, ∂Ω).Hence we can write, using triangle inequality for L p -norms, 2 .In order to complete the proof, in view of eq. ( 14), we need to show that I 2 is bounded as s → 1−.We estimate I 2 in two separate cases.
Case-1: 1 ≤ q ≤ p < ∞.Using Minkowsky's integral inequality and taking the 0-extension of f outside Ω, we have dh ≤ C(p, q, R, N ) f q L p (Ω) .Hence the proof follows in this case.
Case-2: 1 ≤ p ≤ q < ∞.From eq. ( 21) we get that there is some λ f > 0 such that for s ∈ (s 0 , 1), Using Hölder's inequality, we get . Now we can proceed as in Case-1 to show that I 2 ≤ C(N, p, q, R, f ) f p L q (Ω) .This completes the proof.Now we can prove lemma 16 and thereby complete the proof of theorem 1.
Proof of lemma 16.By the standard embedding theorems, we already know that f ∈ L q (Ω).In order to prove the statement, we need to show that when p < q, eq. ( 21) holds.Let Ω 1 be a smooth domain such that {x ∈ Ω | dist(x, ∂Ω) > R} ⊆ Ω 1 ⊆ Ω.