Sharp trace inequalities for fractional Laplacians

The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb's sharp form of the Hardy-Littlewood-Sobolev inequality.


Introduction
In a widely cited paper [4], José F. Escobar proved the trace inequality Here H n is the n-dimensional upper half-space {(x, t) : x ∈ R n , t > 0} and (τ f )(x) denotes the trace of the function f on the boundary of H n . The constant C n is given by and is the sharp constant in (1). Moreover, there is equality in (1) if and only if f is a multiple of the function 1 ((a + t) 2 + |x + b| 2 ) n−2 2 , a > 0 , b ∈ R n .
Escobar observed that the problem is conformally invariant, i.e, the problem can be mapped to the n dimensional sphere using stereographic projection. He then proceeds by showing first that an optimizer exists and then, using the method of Obata, to prove that f 4 n−2 s (s is the standard metric on Euclidean space) is an Einstein metric and hence flat. An entirely different proof of this inequality was found by Beckner [1]. He deduced this inequality from the spectral form Lieb's sharp Hardy-Littlewood-Sobolev inequality on the sphere. Inequality (1) is then recovered via stereographic projection.
Another way of getting Lieb's sharp HLS inequality into the picture was shown in [3]. It is based on the simple and well known observation that among all functions f (x, t) in the upper half space that have a given trace g, the harmonic function By Gauss' theorem A. E. and M. L. were supported in part by NSF grant DMS-0901304. 1 Thus, the inequality (1) takes the form of a Sobolev inequality for fractional derivatives for which Lieb's sharp HLS inequality ( [5]) yields the sharp constant, including all the cases of equality. Another important approach to Escobar's inequality is based on transportation theory [7]. This approach has been generalized in [8] replacing the two norm by general L p -norms. In view of these developments, it is natural to investigate the sharp trace inequalities for fractional Sobolev spaces. More specifically one may ask for the sharp constant for the inequality where As usual, denotes the Fourier transform. There was an attempt in [9], where it was shown that for α ∈ (0, 1) and h defined by (4) is the sharp constant. Using the Fourier transform in the variable x, it is easy to see that the right side of (11) can be written as 2 2α−1 Γ(2 − 2α)(g, (−∆) α g) , and hence inequality (11) appears as a Sobolev inequality for the fractional Laplacian on R n−1 but not as one for the fractional Laplacian on R n . Note that the harmonic function h does not minimize the right side of (11). Minimizing the right side over all functions with a fixed boundary value g yields a function u(x, t) that satisfies the equation The fractional Laplacian is then recovered by the formula where C is some constant. This and general properties of functions that are 'harmonic' in this sense have been derived in [2]. The aim of our little note is to extend the result of Escobar to true fractional Laplacians and prove a class of trace inequalities in their sharp form. This seems to be justified in view of the popularity of Escobar's result and of the simplicity of our proof of its generalization.
It is customary to define the space H α to be the space of all functions in f ∈ L 2 (R n ) whose Fourier transform f satisfies The advantage of this definition is that f , being in L 2 (R n ) is automatically a function. Here we are forced to take another route. For 2α < n we define the space D α (R n ) as the space of tempered distributions in S ′ (R n ) whose Fourier transform is a function in L 2 (R n , |k| 2α dk). More precisely, it consists of elements T ∈ S ′ (R n ) with the property that there exists a function f ∈ L 2 (R n , |k| 2α dk) such that for all φ ∈ S(R n ) It is easy to see that D α endowed with the norm is a Hilbert space. Note, the factor 2π is convenient since we can interpret the right side as It is a standard result that τ 1 extends to a bounded operator from H α (R n ) to H α−1/2 (R n−1 ).
For the case where f ∈ D α (R n ) more can be said. where for some a ∈ R n−m and γ = 0.
Note that for the case α = 1 and m = 1 our constant differs from Esobar's by a factor 1/2. This is due to the fact that our functions are defined on the whole space. For general values of α, D α (H n ) is not defined unless one considers only functions that have support in H n . Also, in the case of α = 1 and m = 1 (15) is easily computable and yields (up to a constant) the optimizer in Escobar's paper [4], as expected.
As mentioned above, trace theorems from H α (R n ) to H α−1/2 (R n−1 ) are standard. Theorem 2.3 in the next section yields the exact norm of the trace as a linear operator from D α (R n ) to D α−1/2 (R n−1 ). This, together with the sharp Sobolev inequality will yield Theorem 1.1. which is, as far as we know, also new for integer α, e.g., α = 2 and n ≥ 5. The tools used in the various proofs, with the exception of Lieb's sharp Hardy-Littlewood-Sobolev inequality, are all elementary.
Acknowledgement: We would like to thank Rupert Frank for helpful discussions concerning this work.

Proof of Theorem 1.1
That f ∈ D α (R n ) is in fact a function follows among other things from the next theorem. Its proof is patterned after the proof of the Sobolev inequality for relativistic kinetic energy |p| in [6] Theorem 8.4., and is just a dual version of Lieb's sharp Hardy-Littlewood-Sobolev inequality.
where s = 2n/(n − 2α). The constant on the right side of (16) is best possible and there is equality if and only if Remark 2.2. As a consequence of the above theorem and the density of S(R n ) in D α (R n ) we have that any distribution in D α (R n ) is a function. Thus, we can say that D α (R n ) is the space of all functions f ∈ L 2n/(n−2α) (R n ) for which there exists a function g ∈ L 2 (R n , |k| 2α dk) such that for all φ ∈ S(R n ). This function g is unique and we shall denote it by f (k).
Proof. For f, g ∈ S(R n ) we have that and by Schwarz's inequality (see Corollary 5.10 in [6]), we may use Lieb's sharp form of the Hardy-Littlewood-Sobolev inequality [5] (see also [6], [3]), which states that R n R n g(x)g(y) |x − y| n−2α dxdy ≤ π n/2−α Γ(α) Γ(n/2 + α) where r = 2n/(n + 2α) < 2. There is equality if and only if where A ∈ C, γ > 0 and a ∈ R n . Thus, Dα g 2 r and taking the supremum over all g ∈ S(R n ) yields inequality (16) for f ∈ S(R n ). Now let f ∈ D α (R n ) be any distribution. Since S(R n ) is dense, there exists a sequence of functions f j ∈ S(R n ) with Thus, f j is a Cauchy sequence in L s (R n ) with a limit which we call φ f . Since f j converges to f in S ′ (R n ) and since the Fourier transform is an isomorphism on S ′ (R n ), we have that f j converges to f in S ′ (R n ). Hence, φ f = f and the inequality (16) is valid for all f ∈ D α (R n ). Similar approximation arguments show that equation (18) continuous to hold for all functions f ∈ D α (R n ) and all g ∈ L r (R n ). Hence, to establish the cases of equality we need equality in (19) which implies that where C is some constant and where g is such that Further, there must be equality in the HLS inequality (21) which mean that g must be of the form (22). This function is integrable and smooth and hence its Fourier transform is bounded with fast decay. Hence (24) also holds for this particular function. It remains to show that f is of the form (17). To see this we imitate the proof of Corollary 5.9 in [6] and obtain that the inverse Fourier transform of g(k)/|k| 2α is given by Since g is an optimizer for the HLS inequality it must satisfy the Euler-Lagrange equation where C is some constant. This yields the form of the optimizer (17).

Theorem 2.3 (Trace inequality).
Assume that m 2 < α < n 2 . Then the trace τ m has a unique extension to a bounded operator τ m : D α (R n ) → D α−m/2 (R n−m ). Moreover, for any f ∈ D α (R n ) The constant in this inequality is best possible and there is equality only if Here k 1 ∈ R n−m and k 2 ∈ R m .
Proof. It is sufficient to prove the statement for m = 1. The rest follows by iterating the result for m = 1. Let f ∈ S(R n ) and note that By Schwarz's inequality, which establishes (25) for all f ∈ S(R n ) and hence the continuous extension of τ 1 to all of D α (R n ). Pick any f ∈ D α (R n ) and let f j ∈ S(R n ) converge to f in D α (R n ). Hence τ 1 f j converges to τ 1 f in D α−1/2 (R n ) and in particular in L 2(n−1)/(n−2α) (R n−1 ) by Theorem 2.1. The formula (τ 1 f j )(k ′ ) = R f j (k ′ , k n )dk n holds for all j. Integrating this against a test function φ ∈ S(R n−1 ) yields Hence, Thus, the steps in (28) are valid for any f ∈ D α (R n ) and the cases of equality follow from the cases of equality in Schwarz's inequality.
Proof of Theorem ??. The inequality follows from using Theorem 2.3 with f and Theorem 2.1 with τ m f . In order to have equality we must have that f satisfies (26) with (27). Also, τ m f must satisfy (23) where g is of the form (22) with the appropriate dimension. Since we conclude that f is of the form (26) with g of the form (22) with the appropriate dimension in the variables. This leads to (15).