Optimal Functional Inequalities for Fractional Operators on the Sphere and Applications

1 Introduction and Main Results

Let us consider the unit sphere 𝕊n with n≥1 and assume that the measure d⁢μ is the uniform probability measure, which is also the measure induced on 𝕊n by Lebesgue’s measure on ℝn+1, up to a normalization constant. With λ∈(0,n), p=2⁢n2⁢n-λ∈(1,2) or equivalently λ=2⁢np′ where 1p+1p′=1, according to [38], the sharp Hardy–Littlewood–Sobolev inequality on 𝕊n reads

For the convenience of the reader, the definitions of all parameters, their ranges and their relations have been collected in Appendix C.

By the Funk–Hecke formula, the left-hand side of the inequality can be written as

where F=∑k=0∞F(k) is a decomposition on spherical harmonics, so that F(k) is a spherical harmonic function of degree k. See [33, Section 4] for details on the computations and, e.g., [42] for further related results. With the above representation, inequality (1.1) is equivalent to

By duality, with q⋆=q⋆⁢(s) defined by

or equivalently s=n⁢(1-2q⋆), we obtain the fractional Sobolev inequality on 𝕊n:

for any s∈(0,n), where

and

With s∈(0,n) the exponent q⋆ is in the range (2,∞). Inequalities (1.1) and (1.5) are related by q⋆=p′ so that

We shall refer to q=q⋆⁢(s) given by (1.4) as the critical case and our purpose is to study the whole range of the subcritical interpolation inequalities

for any q∈[1,2)∪(2,q⋆], where

If q=q⋆, inequalities (1.5) and (1.7) are identical, the optimal constant in (1.7) is 𝖢q⋆,s=κn,sq⋆-2, and we recall that (1.5) is equivalent to the fractional Sobolev inequality on the Euclidean space (see the proof of Theorem 1.6 in Section 3 for details). The usual conformal fractional Laplacian is defined by

For brevity, we shall say that ℒs is the fractional Laplacian of order s, or simply the fractional Laplacian.

We observe that γ0⁢(nq)-1=0 and γ1⁢(nq)-1=q-2. A straightforward computation gives

where the spectrum of ℒs is given by

The case corresponding to s=2 and n≥3, where 1κn,2=14⁢n⁢(n-2), ℒ2=-Δ, 𝒜2=-Δ+14⁢n⁢(n-2), and Δ stands for the Laplace–Beltrami operator on 𝕊n, has been considered by W. Beckner; in [5, p. 233, (35)] he observed that

if q∈(2,q⋆⁢(2)], where q⋆=q⋆⁢(2)=2⁢nn-2 and (k⁢(k+n-1))k∈ℕ is the sequence of the eigenvalues of -Δ according to, e.g., [7]. This establishes the interpolation inequality

where 𝖢q,2=1n is the optimal constant; see [5, (35), Theorem 4] for details. An earlier proof of the inequality with optimal constant can be found in [8, Corollary 6.2], with a proof based on rigidity results for elliptic partial differential equations. Our main result generalizes the interpolation inequalities (1.8) to the case of the fractional operators ℒs, and relies on W. Beckner’s approach. In particular, as in [5], we characterize the optimal constant 𝖢q,s in (1.7) using a spectral gap property.

After dividing both sides of (1.8) by (q-2) we obtain an inequality which, for s=2, also makes sense for any q∈[1,2). When q=1, this is actually a variant of the Poincaré inequality (or, to be precise, the Poincaré inequality written for |F|), and the range q>1 has been studied using the carré du champ method, also known as the Γ2 calculus, by D. Bakry and M. Emery in [3]. Actually their method covers the range corresponding to 1≤q<∞ if n=1 and

In the special case q=2, the left-hand side of (1.8) has to be replaced by the entropy

Still under the condition that s=2, the whole range 1≤q<∞ when n=2, and 1≤q≤2⁢nn-2 if n≥3 can be covered using nonlinear flows as shown in [21, 24, 25].

All these considerations motivate our first result, which generalizes known results for ℒ2=-Δ to the case of the fractional Laplacian ℒs.

With our previous notations, this amounts to

Remarkably, 𝖢q,s is independent of q. Equality in (1.7) is achieved by constant functions. The issue of the optimality of 𝖢q,s is henceforth somewhat subtle. If we define the functional

on the subset ℋs/2 of the functions in Hs/2⁢(𝕊n) which are not almost everywhere constant, then 𝖢q,s can be characterized by

This minimization problem will be discussed in Section 4.

Our key estimate is a simple convexity observation that is stated in Lemma 2.2. The optimality in (1.7) is obtained by performing a linearization, which corresponds to an asymptotic regime as we shall see in Section 2.1. Technically, this is the reason why we are able to identify the optimal constant. The asymptotic regime can be investigated using a flow. Indeed, a first consequence of Theorem 1.1 is that we may apply entropy methods to the generalized fractional heat flow

Notice that (1.10) is a 1-homogeneous equation, but that it is nonlinear when q≠1 and s≠2. Let us define a generalized entropy by

It is straightforward to check that for any positive solution to (1.10) which is smooth enough and has sufficient decay properties as |x|→+∞, we have

so that by applying (1.7) to F=u1q, we obtain the exponential decay of ℰq⁢[u⁢(t,⋅)].

The exponential rate is determined by the asymptotic regime as t→+∞. The value of the optimal constant𝖢q,s is indeed determined by the spectral gap of the linearized problem around non-zero constant functions. From the expression of (1.10), which is not even a linear equation whenever s≠2, we observe that the interplay of optimal fractional inequalities and fractional diffusion flows is not straightforward, while for s=2 the generalized entropy ℰq enters in the framework of the so-called φ-entropies and is well understood in terms of gradient flows; see for instance [2, 13, 28]. When s=2, it is also known from [3] that heat flows can be used in the framework of the carré du champ method to establish the inequalities at least for exponents in the range q≤2# if n≥2, and that the whole subcritical range of exponents can be covered using nonlinear diffusions as in [21, 24, 25] (and also the critical exponent if n≥3). Even better, rigidity results, that is, uniqueness of positive solutions (which are therefore constant functions), follow by this technique. So far there is no analogue in the case of fractional operators, except for one example found in [12] when n=1.

When s=2, the carré du champ method provides us with an integral remainder term and, as a consequence, with an improved version of (1.7). As we shall see, our proof of Theorem 1.1 establishes another improved inequality by construction; see Corollary 2.3. This also suggests another direction, which is more connected with the duality that relates (1.1) and (1.5). Let us describe the main idea. The operator 𝒦s is positive definite and we can henceforth consider 𝒦s1/2 and 𝒦s-1. Moreover, using (1.2) and (1.6), we know that

Expanding the square

with G=Fq⋆-1 so that F⁢G=Fq⋆=Gp where q⋆ and p are Hölder conjugates, we get a comparison of the difference of the two terms which show up in (1.1) and (1.5) and, as a result, an improved fractional Sobolev inequality on 𝕊n. The reader interested in the details of the proof is invited to consult [27] for a similar result.

Still in the critical case q=q⋆, by using the fractional Yamabe flow and taking inspiration from [23, 27, 37, 36, 40], it is possible to give improvements of the above inequality and in particular improve on the constant which relates the left- and the right-hand sides of the inequality in Proposition 1.3. We will not go further in this direction because of the delicate regularity properties of the fractional Yamabe flow and because so far the method does not allow to characterize the best constant in the improvement. Let us mention that, in the critical case q=q⋆, further estimates of Bianchi–Egnell type have also been obtained in [15, 40] for fractional operators. In this paper, we shall rather focus on the subcritical range. It is however clear that there is still space for further improvements, or alternative proofs of (1.5) which rely neither on rearrangements as in [38] nor on inversion symmetry as in [31, 32, 33], for the simple reason that our method fails to provide us with a proof of the Bianchi–Egnell estimates in the critical case.

For completeness let us quote a few other related results. Symmetrization techniques and the method of competing symmetries are both very useful to identify the optimal functions; the interested reader is invited to refer to [39] and [10], respectively, when s=2. In this paper, we shall use notations inspired by [5], but at this point it is worth mentioning that in [5] the emphasis is put on logarithmic Hardy–Littlewood–Sobolev inequalities and their dual counterparts, which are n-dimensional versions of the Moser–Trudinger–Onofri inequalities. Some of these results were obtained simultaneously in [11] with some additional insight on optimal functions gained from rearrangements and from the method of competing symmetries. Concerning observations on duality, we refer to the introduction of [11], which clearly refers the earlier contributions of various authors in this area. For more recent considerations on n-dimensional Moser–Trudinger–Onofri inequalities see, e.g., [19].

Section 2 is devoted to the proof of Theorem 1.1. As already said, we shall take advantage of the subcritical range to obtain remainder terms and improved inequalities. Improvements in the subcritical range have been obtained in the case of non-fractional interpolation inequalities in the context of fast diffusion equations in [29, 30]. In this paper we shall simply take into account the terms which appear by difference in the proof of Theorem 1.1; see Corollary 2.3 in Section 2.3. Although this approach does not provide us with an alternative proof of the optimality of the constant 𝖢q,s in (1.7), variational methods will be applied in Section 4 in order to explain a posteriori why the value of the optimal value of 𝖢q,s is determined by the spectral gap of a linearized problem. Some useful information on the spectrum of ℒs is detailed in Appendix A.

Our next result is devoted to the singular case of inequality (1.7) corresponding to the limit as q=2. We establish a family of sharp fractional logarithmic Sobolev inequalities in the subcritical range.

This result completes the picture of Theorem 1.1 and shows that, under appropriate precautions, the case q=2 can be put in a common picture with the cases corresponding to q≠2. Taking the limit as s→0+, we recover Beckner’s fractional logarithmic Sobolev inequality as stated in [4, 6]. In that case, q=2 is critical from the point of view of the fractional operator. The proof of Corollary 1.4 and further considerations on the s=0 limit will be given in Section 2.4.

Definition (1.6) of 𝒦s also applies to the range s∈(-n,0) and the reader is invited to check that

is defined by the sequence of eigenvalues γk⁢(np) where p=2⁢nn+s is the Hölder conjugate of q⋆⁢(s) given by (1.4). It is then straightforward to check that the sharp Hardy–Littlewood–Sobolev inequality on 𝕊n (see (1.3)) can be written as

where

Notice that κn,-s=1κn,s. A first consequence is that we can rewrite the result of Proposition 1.3 as

for any F∈Hs/2⁢(𝕊n) and G=Fq⋆-1, where n≥1, s∈(0,n), q⋆ is given by (1.4) and p=q⋆′. A second consequence of the above observations is the extension of Theorem 1.1 to the range (-n,0).

The results of Theorems 1.1 and 1.5 are summarized in Figure 1.

Figure 1

The optimal constant 𝖢q,s in (1.7) is independent of q and determined for any given s by the critical case q=q⋆⁢(s) which corresponds to the Hardy–Littlewood–Sobolev inequality (1.1) if s∈(-n,0) and to the Sobolev inequality (1.5) if s∈(0,n). The case s=0 is covered by Corollary 2.4, while q=2 corresponds to the fractional logarithmic Sobolev inequality (2.3) if s=0 and the subcritical fractional logarithmic Sobolev inequality by Corollary 1.4 if s∈(0,n].

To conclude with the outline of this paper, Section 3 is devoted to the stereographic projection and consequences for functional inequalities on the Euclidean space. By stereographic projection, (1.5) becomes

where

and the optimal constant is such that

The fact that (1.5) is equivalent to the fractional Sobolev inequality on the Euclidean space is specific to the critical exponent q=q⋆⁢(s). In the subcritical range weights appear. Let us introduce the weighted norm

The next result is inspired by a non-fractional computation done in [26] and relies on the stereographic projection.

This result is one of the few examples of optimal functional inequalities involving fractional operators on ℝn. It touches the area of fractional Hardy–Sobolev inequalities and weighted fractional Sobolev inequalities, for which we refer to [34, 14] and [16], respectively, and the references therein. The wider family of Caffarelli–Kohn–Nirenberg type inequalities raises additional difficulties, for instance related with symmetry and symmetry breaking issues, which are so far essentially untouched in the framework of fractional operators, up to few exceptions like [14].

Inequality (1.13) holds not only for the space C0∞⁢(ℝn) of all smooth functions with compact support but also for the much larger space of functions obtained by completion of C0∞⁢(ℝn) with respect to the norm defined by

2 Subcritical Interpolation Inequalities

In this section, our purpose is to prove Theorem 1.1.

2.2 Some Spectral Estimates

Let us start with some observations on the function γk in (1.6). Expanding its expression, we get that

γ k ⁢ ( x ) = ( n + k - 1 - x ) ⁢ ( n + k - 2 - x ) ⁢ ⋯ ⁢ ( n - x ) ( k - 1 + x ) ⁢ ( k - 2 + x ) ⁢ ⋯ ⁢ x

for any k≥1. Taking the logarithmic derivative, we find that

(2.2) α k ⁢ ( x ) := - γ k ′ ⁢ ( x ) γ k ⁢ ( x ) = ∑ j = 0 k - 1 β j ⁢ ( x )   with   β j ⁢ ( x ) := 1 n + j - x + 1 j + x ,

and observe that αk is positive. As a consequence, γk′<0 on [0,n] and, from the expression of γk, we read that γk⁢(n)=0. Since γk⁢(n2)=1, we know that γk⁢(nq)>1 if and only if q>2. Using the fact that

γ k ′′ ⁢ ( x ) γ k ⁢ ( x ) = ( α k ⁢ ( x ) ) 2 - α k ′ ⁢ ( x ) = ( γ k ′ ⁢ ( x ) γ k ⁢ ( x ) ) 2 + ∑ j = 0 k - 1 ( 2 ⁢ j + n ) ⁢ ( n - 2 ⁢ x ) ( n + j - x ) 2 ⁢ ( j + x ) 2 ,

we have γk′′⁢(x)≥0, which establishes the convexity of γk on [0,n2]. Moreover, we know that

γ k ′ ⁢ ( n 2 ) = - α k ⁢ ( n 2 ) = - ∑ j = 0 k - 1 4 n + 2 ⁢ j .

See Figure 2. Taking these observations into account, we can state the following result.

Lemma 2.2

Assume that n≥1. With the above notations, the function

q ↦ γ k ⁢ ( n q ) - 1 q - 2

is strictly monotone increasing on (1,∞) for any k≥2.

Figure 2

The functions x↦γk⁢(x) and q↦γk⁢(nq) are both convex, and such that γk⁢(n2)=1.

Proof.

Let us prove that q↦γk⁢(nq) is strictly convex with respect to q for any k≥2. Written in terms of x=nq, it is sufficient to prove that

x ⁢ γ k ′′ + 2 ⁢ γ k ′ > 0   for all ⁢ x ∈ ( 0 , n ) ,

which can also be rewritten as

α k 2 - α k ′ - 2 x ⁢ α k > 0 .

Let us prove this inequality. Using the estimates

α k 2 = ( ∑ j = 0 k - 1 β j ) 2 ≥ 2 ⁢ β 0 ⁢ ∑ j = 1 k - 1 β j + ∑ j = 0 k - 1 β j 2 , β 0 2 - β 0 ′ - 2 x ⁢ β 0 = 0 ,

and

2 ⁢ β 0 ⁢ β j + β j 2 - β j ′ - 2 x ⁢ β j = 2 ⁢ ( n + j ) ⁢ ( n + 2 ⁢ j ) ( n - x ) ⁢ ( n + j - x ) ⁢ ( j + x ) 2

for any j≥1, we actually find that

α k 2 - α k ′ - 2 x ⁢ α k ≥ ∑ j = 1 k - 1 2 ⁢ ( n + j ) ⁢ ( n + 2 ⁢ j ) ( n - x ) ⁢ ( j + n - x ) ⁢ ( j + x ) 2   for all ⁢ k ≥ 2 ,

which concludes the proof. Note that as a byproduct, we also proved the strict convexity of γk for the whole range x∈(0,n). See Figure 2 for a summarization of properties of the spectral functions. ∎

Proof of Theorem 1.1.

We deduce from (1.5) that

∥ F ∥ L q ⁢ ( 𝕊 n ) 2 - ∥ F ∥ L 2 ⁢ ( 𝕊 n ) 2 q - 2 ≤ ∑ k = 1 ∞ γ k ⁢ ( n q ) - 1 q - 2 ⁢ ∫ 𝕊 n | F ( k ) | 2 ⁢ 𝑑 μ

because γ0⁢(x)=1. It follows from Lemma 2.2 that

∥ F ∥ L q ⁢ ( 𝕊 n ) 2 - ∥ F ∥ L 2 ⁢ ( 𝕊 n ) 2 q - 2 ≤ ∑ k = 1 ∞ γ k ⁢ ( n q ⋆ ) - 1 q ⋆ - 2 ⁢ ∫ 𝕊 n | F ( k ) | 2 ⁢ 𝑑 μ

= κ n , s q ⋆ - 2 ⁢ ∑ k = 1 ∞ δ k ⁢ ( n q ⋆ ) ⁢ ∫ 𝕊 n | F ( k ) | 2 ⁢ 𝑑 μ = κ n , s q ⋆ - 2 ⁢ ∫ 𝕊 n F ⁢ ℒ s ⁢ F ⁢ 𝑑 μ .

This proves that 𝖢q,s≤κn,sq⋆-2. The reverse inequality has already been shown in (2.1).∎

Proof of Theorem 1.5.

With s∈(-n,0), it turns out that q⋆ defined by (1.4) is in the range (1,2) and plays the role of p in (1.12). According to Lemma 2.2, the inequality holds with the same constant for any q∈(1,q⋆), and this constant is optimal because of (2.1).∎