Affine fractional $L^p$ Sobolev inequalities

Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb R^n$ are established. The new inequalities are stronger than (and directly imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional versions of the affine $L^p$ Sobolev inequalities of Lutwak, Yang, and Zhang. In addition, affine fractional asymmetric $L^p$ Sobolev inequalities are established.


Introduction
Sharp fractional L 2 Sobolev inequalities are receiving increasing attention in the last decades.They are central in the study of solutions of equations involving the fractional Laplace operator (−∆) 1/2 which arises naturally in many non-local problems such as the stationary form of reaction-diffusion equations [9], the Signorini problem (and its equivalent formulation as the thin obstacle problem) [3], and the Dirichlet-to-Neumann operator of harmonic functions in the half space [29].Also, the general operators (−∆) s for s ∈ (0, 1) arise in stochastic theory, associated with symmetric Levy processes (see [29] and the references therein).Let 0 < s < 1 and 1 ≤ p < n/s.The fractional L p Sobolev inequalities state that |x − y| n+ps dx dy (1) for f ∈ W s,p (R n ), the fractional L p Sobolev space of functions f ∈ L p (R n ) with finite right side in (1) (see, for example, [27]).In general, the optimal constants σ n,p,s and extremal functions are not known (see [6] for a conjecture).Equality is always attained in (1).For p = 1, the extremal functions of (1) are multiples of indicator functions of balls and the constants are explicitly known.The only further known case is p = 2, where the constants σ n,2,s can be obtained by duality from Lieb's sharp Hardy-Littlewood-Sobolev inequalities [18] (see, for example, [10]).The asymptotic behavior of σ n,p,s as s → 1 − was studied in [5].Almgren and Lieb [1] and Frank and Seiringer [12] showed that the extremal functions of (1) are radially symmetric and of constant sign.
By a result of Bourgain, Brezis, and Mironescu [4], for f ∈ W 1,p (R n ), the Sobolev space of L p functions f with weak L p gradient ∇f , where (2) for any η ∈ S n−1 .Here, integration on the unit sphere S n−1 is with respect to the (n − 1)-dimensional Hausdorff measure, ω n is the volume of the n-dimensional unit ball and •, • is the inner product on R n .For p = 1 and p = 2, this allows to deduce the sharp L p Sobolev inequalities from (1) by calculating the limit of σ n,p,s /(1 − s) as s → 1 − .Zhang [32] and Lutwak, Yang, and Zhang [24] obtained the following sharp affine L p Sobolev inequality that is significantly stronger than the classical L p Sobolev inequality: for f ∈ W 1,p (R n ) and 1 < p < n, where the inequality between the first and third terms is the classical L p Sobolev inequality and the optimal constants σ n,p were determined by Aubin [2] and Talenti [30].We have rewritten the explicit constant for the first inequality from [24] using (2).Here Π * p f is the L p polar projection body of f , a convex body associated to f that was introduced with different notation in [24] (see Section 2.5), and | • | is the n-dimensional Lebesgue measure.
The main aim of this paper is to establish affine fractional L p Sobolev inequalities that are stronger than the Euclidean fractional L p Sobolev inequalities from (1) and are fractional counterparts of (3).The case p = 1 was studied in [16], so from now on we let p > 1.
There is equality in the first inequality if and only if f = h s,p •φ for some φ ∈ GL(n), where h s,p is an extremal function of (1).There is equality in the second inequality if f is radially symmetric.
In order to prove Theorem 1, we introduce the s-fractional L p polar projection body Π * ,s p f associated to f , defined as the star-shaped set whose gauge function for ξ ∈ S n−1 is (see Section 3 for details).The affine fractional Sobolev inequality now can be written as Since both sides of (4) are invariant under translations of f , and for volumepreserving linear transformations φ : it follows that ( 4) is an affine inequality.In Theorem 10, we will show that lim which establishes the connection to the L p polar projection bodies introduced by Lutwak, Yang and Zhang [24].
In Section 4 we introduce fractional asymmetric L p polar projection bodies as fractional counterparts of the asymmetric L p polar projection bodies of Haberl and Schuster [14], which in turn are functional versions of the asymmetric L p polar projection bodies of convex bodies introduced in [19].We obtain affine fractional asymmetric L p Sobolev inequalities for non-negative functions that are stronger than the inequalities for the symmetric fractional L p polar projection bodies.
In the proofs of the main results, we use anisotropic fractional Sobolev norms, which were introduced in [20,21] and depend on a star-shaped set K ⊂ R n .In Section 10 we discuss which choice of K (with given volume) gives the minimal fractional Sobolev norm and connect it to the corresponding quest for an optimal L p Sobolev norm solved by Lutwak, Yang, and Zhang [25].

Preliminaries
We collect results on function spaces, Schwarz symmetrization, star-shaped sets, anisotropic Sobolev norms and L p polar projection bodies, that will be used in the following.
2.1.Function spaces.For p ≥ 1 and measurable f : R n → R, let We set {f ≥ t} = {x ∈ R n : f (x) ≥ t} for t ∈ R and use similar notation for level sets, etc.We say that f is non-zero, if {f = 0} has positive measure, and we identify functions that are equal up to a set of measure zero.For p ≥ 1, let Here and below, when we use measurability and related notions, we refer to the n-dimensional Lebesgue measure on R n .For 0 < s < 1 and p ≥ 1, we define the fractional Sobolev space W s,p (R n ) as where ∇f is the weak gradient of f .2.2.Symmetrization.For a set E ⊂ R n , the indicator function 1 E is defined by 1 E (x) = 1 for x ∈ E and 1 E (x) = 0 otherwise.Let E ⊆ R n be a Borel set of finite measure.The Schwarz symmetral of E, denoted by E ⋆ , is the closed centered Euclidean ball with same volume as E.
Let f : R n → R be a non-negative measurable function with super-level sets {f ≥ t} of finite measure.The layer cake formula states that for almost every x ∈ R n and allows us to recover the function from its super-level sets.The Schwarz symmetral of f , denoted by f ⋆ , is defined by Hence, f ⋆ is determined by the properties of being radially symmetric, decreasing and having super-level sets of the same measure as those of f .Note that f ⋆ is also called the symmetric decreasing rearrangement of f .
The proofs of our results make use of the Riesz rearrangement inequality, which is stated in full generality, for example, in [7].
Theorem 2 (Riesz's rearrangement inequality).For f, g, k : R n → R non-negative, measurable functions with super-level sets of finite measure, We will use the characterization of equality cases of the Riesz rearrangement inequality due to Burchard [8].
Theorem 3 (Burchard).Let A, B and C be sets of finite positive measure in R n and denote by α, β and γ the radii of their Schwarz symmetrals A ⋆ , B ⋆ and C ⋆ .For |α − β| < γ < α + β, there is equality in

2.3.
Star-shaped sets and star bodies.A set K ⊆ R n is star-shaped (with respect to the origin), if the interval [0, x] ⊂ K for every x ∈ K.The gauge function • K : R n → [0, ∞] of a star-shaped set is defined as and the radial function The n-dimensional Lebesgue measure or volume of a star-shaped set K in R n with measurable radial function is given by We call a star-shaped set K ⊂ R n a star body if its radial function is strictly positive and continuous in R n \ {0}.On the set of star bodies, the q-radial sum for q = 0 of K, L ⊂ R n is defined by for ξ ∈ S n−1 (cf.[28,Section 9.3]).The dual Brunn-Minkowski inequality (cf.[28, (9.41)]) states that for star bodies K, L ⊂ R n and q > 0, with equality precisely if K and L are dilates, that is, there is λ > 0 such that K = λL.
Let α ∈ R\{0, n}.For star-shaped sets K, L ⊆ R n with measurable radial functions, the dual mixed volume is defined as for star-shaped sets K, L 1 , L 2 ⊆ R n with measurable radial functions.
For star-shaped sets K, L ⊆ R n of finite volume and 0 < α < n, the dual mixed volume inequality states that Equality holds if and only if K and L are dilates, where we say that star-shaped sets K and L are dilates if ρ K = λ ρ L almost everywhere on S n−1 for some λ > 0.
The definition of dual mixed volume for star bodies is due to Lutwak [22], where also the dual mixed volume inequality is derived from Hölder's inequality (also see [28,Section 9.3] or [13, B.29]).

Anisotropic fractional
It was introduced in [21] for K a convex body (also, see [20]).For K = B n , the Euclidean unit ball, we obtain the classical s-fractional L p Sobolev norm of f .The limit as s → 1 − was determined in [4] in the Euclidean case and in [21] in the anisotropic case.We will also consider the following asymmetric versions of (7), where a + = max{a, 0} and a − = max{−a, 0} for a ∈ R. The limits as s → 1 − were determined in [26].
2.5.L p polar projection bodies.For p ≥ 1 and f ∈ W 1,p (R n ), the L p polar projection body is defined as the star body with gauge function given by for ξ ∈ S n−1 , were •, • denotes the inner product.It is the polar body of a convex body.The definition is due to Lutwak, Yang, and Zhang [24].For a convex body K ⊂ R n , they defined the L p polar projection body (with a different normalization) in [23] by where S p (K, •) is the L p surface area measure of K (for the definition of L p surface area measures, see, for example, [28, Section 9.1]).
Asymmetric L p polar projection bodies of convex bodies were introduced in [19].For f ∈ W 1,p (R n ), the asymmetric L p polar projection bodies of f are defined as the star bodies with gauge function given by

Fractional L p Polar Projection Bodies
Let 0 < s < 1 and 1 < p < n/s.For f ∈ W s,p (R n ), define the s-fractional L p polar projection body Π * ,s p f as the star-shaped set given by the gauge function p f is a one-homogeneous function on R n .Let K ⊂ R n be a star body.The following simple calculation turns out to be useful.For Next, we establish basic properties of fractional L p polar projection bodies.
Proposition 4. For non-zero f ∈ W s,p (R n ), the set Π * ,s p f is an origin-symmetric star body with the origin in its interior.Moreover, there is c > 0 depending only on f and p such that Π * ,s p f ⊆ c B n for every s ∈ (0, 1).
Proof.First, note that since for ξ ∈ R n and t > 0, Next, we show that Π * ,s p f is bounded.We take r > 1 large enough so that f L p (rB n ) ≥ 2 3 f p and easily see that for t > 2r, Hence, which implies that Π * ,s p f ⊆ c B n for c > 0 independent of s.Now, we show that Π * ,s p f has the origin in its interior.First observe that for ξ, η ∈ R n , by the triangle inequality and a change of variables, Using the relation (10) with K = B n , we get p f < r} has positive (n − 1)-dimensional Hausdorff measure and contains a basis {ξ 1 , . . ., ξ n } ⊆ A of R n .Applying (if necessary) a linear transformation to Π * ,s p f , we may assume without loss of generality that ξ i = e i are the canonical basis vectors.For every x ∈ R n , writing x = x i e i and using (11), we get (12) x where d > 0 is independent of x.This shows that Π * ,s p f has the origin as interior point.
We obtain Applying inequality (13) to the vectors ξ + η and −η, we get The continuity of • Π * ,s p f now follows from ( 13) and ( 14). 4. Fractional Asymmetric L p Polar Projection Bodies Let 0 < s < 1 and 1 < p < n/s.For f ∈ W s,p (R n ), define the asymmetric s-fractional L p polar projection bodies Π * ,s p,+ f and Π * ,s p,− f as the star-shaped sets given by the gauge functions p,+ f and state our results just for Π * ,s p,+ f .Note that, as in the symmetric case, Let K ⊂ R n be a star body and f ∈ W s,p (R n ).As in (10), we obtain that ( 16) In the following proposition, we derive the basic properties of fractional asymmetric L p polar projection bodies.
Proposition 5.For non-zero f ∈ W s,p (R n ), the set Π * ,s p,+ f is a star body with the origin in its interior.Moreover, there is c > 0 depending only on f and p such that Π * ,s p,+ f ⊆ c B n for every s ∈ (0, 1).
Proof.Since the functions (a) p + and (a) p − are convex, the inequalities (a + b) + dx, and take r > 0 so large that R n \rB n |f (x)| p dx < ε.For z ∈ R n \ 2rB n , we obtain by Hölder's inequality that In case R n (f (x)) p + dx = 0 the previous inequality holds trivially for any r > 0. By an analogous calculation, and eventually increasing the value of r, we obtain that Note that Π * ,s p f ⊂ Π * ,s p,+ f .Hence, it follows from Proposition 4 that Π * ,s p,+ f contains the origin in its interior, that is, there is d > 0 such that (17) x Π * ,s p,+ f ≤ d |x| for every x ∈ R n .

The Limit of Fractional L p Polar Projection Bodies
We establish the limiting behavior of s-fractional L p polar projection bodies for 1 < p < n/s as s → 1 − in the symmetric and asymmetric case.For p = 1, a corresponding result was proved in [16].Let 0 < s < 1 and 1 < p < n/s.Set p ′ = p/(p − 1).We say that We require the following lemmas.Lemma 6.The following statements hold.
(1) For f ∈ L p (R n ), ( Proof.First we prove (1).Let g ∈ B p ′ ,+ and write f = f + − f − .Since f − and g are non-negative, it follows from Hölder's inequality that For the opposite inequality, take and notice that g ∈ B p ′ ,+ and Next we prove (2).Fix k 0 and g 0 ∈ B p ′ ,+ .By (1), we have Since this inequality holds for every k 0 , Thus, by (1), Finally, we prove (3).Take c ≥ max{ and the statement follows.
Proof.Let g : R n → R be a smooth function with compact support.Write div x for the divergence taken with respect to the variable x.Using integration by parts, we obtain for ξ ∈ S n−1 and t > 0, t converges weakly to ∇f (•), ξ as t → 0. By Lemma 6 (2), lim inf For the opposite inequality we recall that for any g ∈ B p ′ ,+ , the function x → 1 0 g(x − rtξ) dr is in B p ′ ,+ as well.Hence, Again by Lemma 6 (1), The following result is Lemma 4 in [16].
We are now able to prove the main result of this section. and By Proposition 4 we can use the dominated convergence theorem to obtain lim The following result is an immediate consequence of Theorem 9 and (15).
for every star body K ⊂ R n .

Anisotropic Fractional Pólya-Szegő Inequalities
We will establish anisotropic Pólya-Szegő inequalities for fractional L p Sobolev norms and their asymmetric counterparts.
where k t (z) = 1 t −1/(n+ps) K (z), we obtain Hence, for t > 0, it follows from Fubini's theorem that Clearly the first term is invariant under Schwarz symmetrization.
For the second term, by the Riesz rearrangement inequality, Theorem 2, we have and that the corresponding equation holds for f ⋆ .Hence, if there is equality in (21), then, for (r, r, t) ∈ (0, ∞) 3 \M with |M | = 0, we have For almost every (r, r) ∈ (0, ∞) 2 , we have (r, r, t) ∈ (0, ∞) 3 \M for almost every t > 0. For such (r, r) with r ≤ r and t > 0 sufficiently large, the assumptions of Theorem 3 are fulfilled and therefore there are a centered ellipsoid D and a, b ∈ R n (depending on (r, r, t)) such that {f ≥ r} = a + αD, t −1/(n+ps) K = b + βD, {f ≥ r} = c + γD where c = a + b.Since K = t 1/(n+ps) b + (|K|/|D|) 1/n D, the centered ellipsoid D does not depend on (r, r, t) and also a, c do not depend on t.It follows that b = 0 and that K is a multiple of D. Hence, a = c is a constant vector which concludes the proof.
The following result is a variation of [17,Theorem 3.1].
Equality holds for non-zero f ∈ W s,p (R n ) if and only if K is a centered ellipsoid and f is a translate of f ⋆ • φ for some φ ∈ SL(n). Proof.Since dx dy, the result follows from Theorem 11 for K and −K.

Affine Fractional Pólya-Szegő Inequalities
We establish affine Pólya-Szegő inequalities for fractional asymmetric and symmetric L p polar projection bodies.
Proof.By Theorem 11, ( 16) and the dual mixed volume inequality, we obtain for which completes the proof of the inequality.By Theorem 11, there is equality in (22) if and only if f is a translate of f ⋆ • φ for some φ ∈ SL(n).
The following result is obtained in the same way as Theorem 13 by replacing Theorem 11 with Theorem 12.
We remark that by Theorem 10 we obtain from Theorem 14 in the limit as , which is equivalent to the Pólya-Szegő inequality for L p projection bodies by Cianchi, Lutwak, Yang, and Zhang [11,Theorem 2.1].Similarly, by Theorem 9 we obtain from Theorem 13 in the limit as s → 1 − that , which is equivalent to the Pólya-Szegő inequality for asymmetric L p projection bodies by Haberl, Schuster and Xiao [15, Theorem 1].

Affine Fractional Asymmetric L p Sobolev Inequalities
We establish the following affine fractional asymmetric L p Sobolev inequalities and show that they are stronger than Theorem 1.
Theorem 15.Let 0 < s < 1 and 1 < p < n/s.For non-negative f ∈ W s,p (R n ), There is equality in the first inequality if and only if f = h s,p •φ for some φ ∈ GL(n) where h s,p is an extremal function of (1).There is equality in the second inequality if f is radially symmetric.
Proof.By Theorem 13, The fractional Sobolev inequality (1) shows that Combining these inequalities and their equality cases, we complete the proof of the first inequality of the theorem.
For the second inequality, we set K = B n in ( 16) and apply the dual mixed volume inequality (6) to obtain There is equality precisely if Π * ,s p,+ f is a ball, which is the case for radially symmetric functions.
Note that it follows from the definition of fractional symmetric and asymmetric L p polar projection bodies that Π * ,s p f = Π * ,s p,+ f +−ps Π * ,s p,− f.We use the dual Brunn-Minkowski inequality ( 5) and obtain that with equality precisely if the star bodies Π * ,s p,+ f and Π * ,s p,− f are dilates.Thus, it follows that for non-negative f , Theorem 15 implies Theorem 1 and it is, in general, substantially stronger than Theorem 1.Of course, they coincide for even functions.9. Affine Fractional L p Sobolev Inequalities: Proof of Theorem 1 For non-negative f , the first inequality in Theorem 1 follows from Theorem 15, as mentioned before.For general f and x, y ∈ R n , we use where equality holds if and only if f (x) and f (y) are both non-negative or nonpositive.We obtain with equality if and only if f has constant sign for almost every x, y ∈ R n .Using the result for |f |, we obtain the first inequality of the theorem and its equality case.
For the second inequality, we set K = B n in (10) and apply the dual mixed volume inequality (6) as in the proof of Theorem 15.

Optimal Fractional L p Sobolev Bodies
The following important question was asked by Lutwak, Yang and Zhang [25] for a given f ∈ W 1,p (R n ) and 1 ≤ p < n: For which origin-symmetric convex bodies attained?An optimal L p Sobolev body of f is a convex body where the infimum is attained.Lutwak, Yang ang Zhang [25] showed that the infimum in ( 23) is attained (up to normalization) at the unique origin-symmetric convex body f p in R n such that (24) S n−1 g(ξ) dS p ( f p , ξ) = R n g(∇f (x)) dx for every even g ∈ C(R n ) that is positively homogeneous of degree p, where S p (K, •) is the L p surface area measure of K. Setting g = • K * , they obtain from the L p Minkowski inequality that (25) 1 n R n ∇f (x) p K * dx = V p ( f p , K) ≥ | f p | (n−p)/n |K| p/n , with equality precisely if K and f p are homothetic (see [28,Section 9.1] for the definition of the L p mixed volume V p (•, •) and the L p Minkowski inequality).Hence, they obtain from their solution to their functional version (24) of the L p Minkowski problem that f p is the optimal L p Sobolev body associated to f .Tuo Wang [31] obtained corresponding results for f ∈ BV (R n ) and p = 1.Let 0 < s < 1 and 1 < p < n/s.The results by Lutwak, Yang and Zhang [25] suggest the following question for a given f ∈ W s,p (R n ): For which star bodies L ⊂ R n is (26) inf and there is equality precisely if L is a dilate of Π * ,s p f .Hence, Π * ,s p f is the unique optimal s-fractional L p Sobolev body associated to f .
To understand how the solutions to (23) and ( 26) are related, we use the following result: For f ∈ W 1,p (R n ) and L ⊂ R n a star body, (27) lim | ξ, η | p ρ L (η) n+p dη, is a multiple of the L p centroid body of L. This can be proved as in [21], where the corresponding result was established for a convex body L (with a different normalization of Z p L).It also follows from Theorem 10.Indeed, by (10) and (20), Using that (28) Π * p f = Π * p f p for f ∈ W 1,p (R n ), which follows from (24) by setting g = | •, η | p for η ∈ S n−1 and using ( 8) and (9) (cf.[25]), and that (29) V p (K, Z p L) = Ṽ−p (L, Π * p K) for K a convex body and L a star body, a well-known relation that follows from Fubini's theorem, we now obtain (27) from the first equation in (25).
Using (27), we obtain from (26) in the limit as s → 1 − for a given f ∈ W 1,p (R n ), the following question: For which star bodies L ⊂ R n is (30) inf with equality precisely if L and Π * p f are dilates, where we have used ( 28) and ( 29).From Theorem 10, we obtain that a suitably scaled sequence of optimal s-fractional Sobolev bodies converges to a multiple of the optimal body for (30) as s → 1 − .
if and only if, up to sets of measure zero, A = a + αD, B = b + βD, C = c + γD, where D is a centered ellipsoid, and a, b and c = a + b are vectors in R n .

s→1 − p( 1
− s) R n R n |f (x) − f (y)| p x − y n+ps L dx dy = R n ∇f (x) Z * p L dx,where the convex body Z p K, defined for ξ ∈ S n−1 by h ZpL (ξ) p = S n−1