Fractional Gagliardo-Nirenberg interpolation inequality and bounded mean oscillation

We prove Gagliardo-Nirenberg interpolation inequalities estimating the Sobolev semi-norm in terms of the bounded mean oscillation semi-norm and a Sobolev semi-norm, with some of the Sobolev semi-norms having fractional order.


Introduction
The homogeneous Gagliardo-Nirenberg interpolation inequality for Sobolev space states that if d ∈ N \ {0} and if 0 ≤ s 0 < s < s 1 , 1 ≤ p, p 0 , p 1 ≤ ∞ and 0 < θ < 1 fulfil the condition unless s 1 is an integer, p 1 = 1 and s 1 − s 0 ≤ 1 − 1 p 0 .When s = 0, we use the convention that Ẇ 0,p (R d ) = L p (R d ), and when s ∈ N \ {0} is a positive integer, Ẇ s,p (R d ) is the classical integer-order homogeneous Sobolev space of s times weakly differentiable functions f : R d → R such that D s f ∈ L p (R d ) and .
For s 0 , s 1 , s ∈ N the inequality (1.2) was proved by Gagliardo [15] and Nirenberg [26] (see also [14]).When s ∈ N, the homogeneous fractional Sobolev-Slobodeckiȋ space Ẇ s,p (R d ) can be defined as the set of measurable functions f : R d → R which are k times weakly differentiable with a finite Gagliardo semi-norm: with k ∈ N, σ ∈ (0, 1) and s = k + σ; the characterisation of the range in which the Gagliardo-Nirenberg interpolation inequality (1.2) holds was performed in a series of works [4,[9][10][11] up to the final complete settlement by Brezis and Mironescu [5].We focus on the endpoint case where s 0 = 0 and p 0 = ∞.In this case, the inequality (1.2) becomes and holds under the assumption that sp = s 1 p 1 and either s 1 = 1 or p 1 > 1.It is natural to ask whether the inequality (1.5) can be strengthened by replacing the uniform norm • L ∞ (R d ) by John and Nirenberg's bounded mean oscillation (BMO) semi-norm , which plays an important role in harmonic analysis, calculus of variations and partial differential equations [18], that is, whether we have the inequality where the bounded mean oscillation semi-norm The estimate (1.6) was proved indeed when s = 1, p = 4, s 1 = 2 and p 1 = 2 via a Littlewood-Paley decomposition by Meyer and Rivière [24, theorem 1.4], and for s, s 1 ∈ N via the duality between BMO(R d ) and the real Hardy space H 1 (R d ) by Strezelecki [28]; a direct proof was been given recently by Miyazaki [25] (in the limiting case s 0 = s 1 = 0, see [8; 20, theorem 2.2]); when s 1 < 1, the estimate (1.6) has been proved by Brezis and Mironescu through a Littlewood-Paley decomposition [6, lemma 15.7] (see also [2,20] for similar estimates in Riesz potential spaces).
The main result (theorem 1) of the present work is the estimate (1.6) when s 1 = 1 and 0 < s < 1, with a proof which is quite elementary: the main analytical tool is the classical maximal function theorem.We also show how the same ideas can be used to give a direct proof of (1.6) when s 1 < 1, depending only on the definitions of the Gagliardo and bounded mean oscillation semi-norms (theorem 7).Finally, we show how a last interpolation result (theorem 10) allows one to obtain the full range of interpolation between BMO(R d ) and higher-order fractional Sobolev-Slobodeckiȋ spaces Ẇ s,p (R d ) with s ∈ (1, ∞).
Our proofs can be considered as fractional counterparts of Miyazaki's direct proof in the integer-order case [25].We also refer to Dao's recent work [12] for an alternative approach via negative-order Besov spaces to the results in the present paper.

Interpolation between first-order Sobolev semi-norm and mean oscillation
We prove the following interpolation inequality between the fist-order Sobolev seminorm and the mean oscillation seminorm into fractional Sobolev spaces.Theorem 1.For every d ∈ N\{0} and every p ∈ (1, ∞), there exists a constant C(p) > 0 such that for every s ∈ (1/p, 1), every open convex set Ω ⊆ R d satisfying κ(Ω) < ∞ and every function f ∈ Ẇ 1,sp (Ω) ∩ BMO(Ω), one has f ∈ Ẇ s,p (Ω) and We define here for a domain Ω ⊆ R d , the bounded mean oscillation semi-norm of a measurable function f : Ω → R as and the geometric quantity x ∈ Ω and r ∈ (0, diam(Ω)) .
For the latter quantity, one has for example The quantity κ(Ω) can be infinite for some unbounded convex sets such as Ω = (0, 1) × Our first tool to prove theorem 1 is an estimate by the maximal function of the derivative of the average distance of values on a ball to a fixed value; this formula is related to the Lusin-Lipschitz inequality [1,lemma II.1;3;16,p. 404;17,(3.3 , then for every r ∈ (0, diam(Ω)) and almost every x ∈ Ω, (2.7) Here Mg : R d → [0, +∞] denotes the classical Hardy-Littlewood maximal function of the function g Proof of lemma 2. Since Ω is convex and f ∈ Ẇ 1,1 (Ω), for almost every x ∈ Ω and every r ∈ (0, ∞), we have (2.9) By convexity of the set Ω, for every z ∈ Ω ∩ B r (x) and t ∈ [0, 1] we have (1 − t)x + tz ∈ Ω∩B tr (x).We deduce from (2.8) and (2.9) through the change of variable y = (1−t)x+tz that in view of the definition (2.2) of the maximal function, and the conclusion (2.7) then follows from the definition of the geometric quantity κ(Ω) in (2.3).
Our second tool to prove theorem 1 is the following property of averages of functions of bounded mean oscillation (see [7, §3]).
In (2.11), e denotes Euler's number.The proof of lemma 3 will use the following triangle inequality for averages Lemma 4. Let Ω ⊆ R d .If the function f : Ω → R is measurable, and the sets A, B, C ⊆ R d are measurable and have positive measure, then Proof.We have successively, in view of the triangle inequality, Proof of lemma 3. We first note that since r 1 > r 0 , we have in view of (2.2) since by convexity Applying k ∈ N \ {0} times the inequality (2.12), we get thanks to the triangle inequality for mean oscillation of lemma 4, (2.13) Taking k ∈ N\{0} such that k −1 < d ln(r 1 /r 0 ) ≤ k, we obtain the conclusion (2.11).
Our last tool to prove theorem 1 is the following integral identity.
Lemma 5.For every p ∈ (1, ∞) and α ∈ (0, ∞), one has Proof.One performs the change of variable r = exp(t/α) in the left-hand side integral and uses the classical integral definition of the Gamma function.
We now proceed to the proof of theorem 1.
Proof of theorem 1.For every x, y ∈ Ω, we have by the triangle inequality and the domain monotonicity of the integral (2.15) If ̺ ∈ (0, diam(Ω)), we first have by lemma 2, for almost every x ∈ Ω, (2.16) Next we have by the triangle inequality, by lemma 2 again and by lemma 3, for every r ∈ (̺, diam(Ω)), (2.17) and hence, integrating (2.17), we get in view of lemma 5. Putting (2.16) and (2.18) together, we get, since sp > 1, We conclude this section by pointing out that theorem 1 admits a localised version in terms of Fefferman and Stein's sharp maximal function f ♯ : Ω → [0, ∞] which is defined for every x ∈ Ω (see [13, (4 Proposition 6 is stronger than theorem 1 in the sense that the integration of the estimate (2.23) yields (2.1).
Proposition 6 is a counterpart of the interpolation involving maximal and sharp maximal function of derivatives [22, (4)], which generalised a priori estimates in terms of maximal functions [19; 23, theorem 1]; proposition 6 generalises the corresponding result for integer-order Sobolev spaces [25, remark 2.2].

Interpolation between first-order Sobolev semi-norm and mean oscillation
We explain how the tools of the previous section can be used to prove the fractional BMO Gagliardo-Nirenberg interpolation inequality as persented by Brezis and Mironescu's [6, lemma 15.7].
Theorem 7.For every d ∈ N \ {0}, every s, s 1 ∈ (0, 1) and every p, p 1 ∈ (1, +∞) satisfying s < s 1 and s 1 p 1 = sp, there exists a constant C > 0 such that for every open convex set Ω ⊆ R d satisfying κ(Ω) < ∞ and for every function f ∈ Ẇ s 1 ,p 1 (Ω)∩BMO(Ω), one has f ∈ Ẇ s,p (Ω) and The proof of theorem 7 will follow essentially the proof of theorem 1, the main difference being the replacement of lemma 2 by its easier fractional counterpart.

Lemma 8. For every
Proof.By Hölder's inequality we have for every r ∈ (0, diam(Ω)) and for every x ∈ Ω, we reach the conclusion (3.2) thanks to the definition of the geometric quantity κ(Ω) in (2.3).
Proof of theorem 7. We begin as in the proof of theorem 1.Instead of (2.16), we have by lemma 8, We conclude by integration of (3.8).
As previously, we point out that the estimate (3.8) admits a localised version, which is the fractional counterpart of proposition 6. Proposition 9.For every d ∈ N \ {0}, every s, s 1 ∈ (0, 1) and every p, p 1 ∈ (1, +∞) satisfying s < s 1 and s 1 p 1 = sp, there exists a constant C > 0 such that for every open convex set Ω ⊆ R d satisfying κ(Ω) < ∞, for every measurable function f : Ω → R and for every x ∈ Ω, (3.9) ˆdiam(Ω) The estimate (3.1) can be seen as a consequence of the integration of (3.9).

Higher-order fractional spaces estimates
The last ingredient to obtain the full scale of Gagliardo-Nirenberg interpolation inequalities between fractional Sobolev-Slobodeckiȋ spaces and the bounded mean oscillation space is the following estimate.

1 p 1 .( 4 . 4 )D k 1 f
For the second-term in the right-hand side of (4.3), for every x ∈ R d , we have by weak differentiability(y) dy ≤C 15 ̺ k 1 f BMO(R d ) .