In this paper, some sharp Sobolev inequalities on BV(Rn), the space of functions of bounded variation on Rn, n≥2, are deduced through the Lp Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on BV(Rn).
Citation: Jin Dai, Shuang Mou. Some sharp Sobolev inequalities on BV(Rn)[J]. AIMS Mathematics, 2022, 7(9): 16851-16868. doi: 10.3934/math.2022925
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In this paper, some sharp Sobolev inequalities on BV(Rn), the space of functions of bounded variation on Rn, n≥2, are deduced through the Lp Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on BV(Rn).
The sharp Sobolev inequality on the Sobolev space W1,1(Rn), n≥2, going back to [10,11,30], states that
nω1/nn‖f‖1∗≤‖|∇f|‖1, |
where 1∗=n/(n−1), |∇f| is the Euclidean norm of the weak gradient of f, ‖f‖p is the usual Lp norm of f in Rn and ωk is the volume enclosed by the unit sphere Sk−1 in Rk. It is one of the fundamental inequalities in many branches of analysis and geometry. So far, the Sobolev inequality and relatives of the Sobolev inequality have been investigated intensively (see, for example, [1,2,3,4,5,6,7,8,17,21,24,27,31,37]).
Cianchi-Talenti [6,31] established an extended version of the Sobolev inequality on BV(Rn), and the equality is actually attained. The relevant inequality states that if f∈BV(Rn), then we have
nω1/nn‖f‖1∗≤‖Df‖, | (1.1) |
where ‖Df‖ is the total variation of f and equality holds if and only if f is a multiple of the characteristic function of some ball.
In this work, for p>1, the family of sharp Sobolev inequalities is established, which reads as follows:
∫Sn−1(∫Rn|u⋅σf|pd|Df|)−npdu≤(1n)n−1(1cn−2,p)np‖f‖−n1∗‖Df‖n(p−1)p, | (1.2) |
for f∈BV(Rn). Here the symbol ⋅ is the standard Euclidean scalar product, du is the standard spherical Lebesgue measure, the vector valued Radon measure Df is the weak gradient of f, |Df| is the variation measure of f and σf is the Radon-Nikodym derivative of Df with respect to |Df| (cf. Section 3). On the other hand, through the Lp Brunn-Minkowski theory, we prove that these inequalities can all imply the sharp Sobolev inequality (1.1). In fact, this is a direct consequence of the H¨older inequality and the Lp Cauchy surface area (cf. Section 5).
In [38], Zhang proved the affine Sobolev-Zhang inequality on C1c(Rn), the space of C1 functions of compact support on Rn, n≥2, which states
1n∫Sn−1‖u⋅∇f‖−n1du≤(ωn2ωn−1)n‖f‖−n1∗. | (1.3) |
While Zhang showed that the inequality (1.3) is stronger than the sharp Sobolev inequality on C1c(Rn) and is equivalent to the Petty projection inequality, which is a famous affine inequality in convex geometry and directly implies the isoperimetric inequality for convex bodies.
The inequality (1.3) is sharp, although equality is not attained on C1c(Rn) unless f=0 a.e. with respect to the Lebesgue measure on Rn (simply write f=0 for this in the paper). But the characteristic functions of ellipsoids can be considered to be the virtual extremals. In [33], Wang proved an extended version of the affine Sobolev-Zhang inequality on the space of functions of bounded variation on Rn, BV(Rn), and these characteristic functions turn into actual extremals. The extended inequality states that for f∈BV(Rn),
1n∫Sn−1(∫Rn|u⋅σf|d|Df|)−ndu≤(ωn2ωn−1)n‖f‖−n1∗. | (1.4) |
It is proved that equality holds if and only if f=λ1E for some λ∈R and some ellipsoid E⊂Rn. Here 1A denotes the characteristic function of A⊂Rn.
Analogously, the extended affine Sobolev-Zhang inequality is stronger than the sharp Sobolev inequality on BV(Rn). And the inequality (1.4) is GL(n) invariant while the Sobolev inequality is O(n) invariant, where GL(n) and O(n) denote the general linear group and the orthogonal transformation group on Rn, respectively. Particularly, Wang deduced the Petty projection inequality for sets of finite perimeter, which directly implies the isoperimetric inequality for sets of finite perimeter (see [33]).
In [27], Lutwak, Yang and Zhang had extended (1.3) to sharp affine Lp Sobolev inequalities on Sobolev spaces W1,p(Rn), 1<p<n. The Sobolev affine energy of f is defined as
Ep(f)=(1n∫Sn−1‖u⋅∇f‖−npdu)−1n, |
for 1<p<n. About the related research of the energy Ep, for example, please see [14,15,20,22,23].
If μ is an outer measure on Rn, f:Rn→[0,∞] is μ-measurable and E⊂Rn is a μ-measurable set with μ(E)<∞, then we set
−∫Efdμ=1μ(E)∫Efdμ. |
Now assuming f≠0, we can rewrite the inequality (1.4) as
1n∫Sn−1( −∫Rn|u⋅σf|d|Df|)−ndu≤(ωn2ωn−1)n(‖Df‖‖f‖1∗)n. | (1.5) |
For p>1, using the H¨older inequality, we have
−∫Rn|u⋅σf|d|Df|≤( −∫Rn|u⋅σf|pd|Df|)1p, |
where equality holds if and only if |u⋅σf|=constant. Then (1.5) implies
1n∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu≤(ωn2ωn−1)n(‖Df‖‖f‖1∗)n. | (1.6) |
But the inequality (1.6) is not sharp. The equality in (1.6) is never attained. In fact, λ1E is the only extremal in (1.5) for any λ∈R and any ellipsoid E, while |u⋅σλ1E| is never equal to some constant indentically.
So, in this paper, we refine the constant (ωn/(2ωn−1))n in the inequality (1.6). We prove the sharp form of (1.6) (or the inequality (1.2)) in the following.
Theorem 1.1. Let f∈BV(Rn), f≠0 and p>1. Then
∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu≤(1n)n−1(1cn−2,p)np(‖Df‖‖f‖1∗)n, | (1.7) |
where equality holds whenever f=λ1B for some λ∈R and some ball B⊂Rn. Moreover, for p≠n, if the equality holds in (1.7), then f=λ1B for some λ∈R and some ball B⊂Rn.
Here
cn−2,p=ωn+p−2ω2ωn−2ωp−1, | (1.8) |
ωs=πs/2/Γ(1+s/2) for s≥0 and Γ(⋅) is the Gamma function. Note that if p→1+ in (1.7), we can get the affine Sobolev inequality (1.5).
Like the extended affine Sobolev-Zhang inequality, the sharp inequality (1.7) can imply the sharp Sobolev inequality on BV(Rn) (cf. Section 5). However, these inequalities are only O(n) invariant rather than G(n) invariant.
Throughout the paper f is not equal to 0, a.e., with respect to the Lebesgue measure on Rn unless we give the particular remark.
In this paper, the main tool is the Lp Brunn-Minkowski theory in convex geometry. We will use the method of the convexification which has been used in [29,33,38]. For example, in [29], Lutwak-Yang-Zhang associated with each Sobolev function f∈W1,1(Rn) an origin-symmetric convex body (compact convex set with non-empty interior) ⟨f⟩ by using the even Minkowski problem, which reads as follows: Given f∈W1,1(Rn), there exists a unique origin-symmetric convex body ⟨f⟩ such that
∫Sn−1g(u)dS(⟨f⟩,u)=∫Rng(∇f(x))dx | (1.9) |
for all even continuous functions g on Rn that are positively homogeneous of degree 1. Here S(⟨f⟩,⋅) is the Alexandrov-Fenchel-Jessen surface area measure of ⟨f⟩ (cf. Section 2). We call ⟨f⟩ the Lutwak-Yang-Zhang (LYZ) body of f and call Φ:W1,1(Rn)→Kne (see below for its definition), Φ(f)=⟨f⟩, the LYZ operator. It is shown in [33] that the LYZ operator can be extended to BV(Rn): Given f∈BV(Rn), there exists a unique origin-symmetric convex body ⟨f⟩ such that
∫Sn−1g(u)dS(⟨f⟩,u)=∫Rng(σf)d|Df| | (1.10) |
for all even continuous functions g on Rn that are positively homogeneous of degree 1.
In recent years, the LYZ operator ⟨⋅⟩ has been studied and used widely in convex geometry and other mathematical areas. For example, Ludwig [25] established that each continuous affinely covariant Blaschke valuation on W1,1(Rn) is the LYZ operator ⟨⋅⟩. For more related information, please see [16,34].
We state two key steps in order to prove Theorem 1.1. On the one hand, we will make use of the LYZ operator on BV(Rn) and the even Lp Minkowski problem. We associate each orgin-symmetric convex body K with the another orgin-symmetric convex body: Given p>1, p≠n and K∈Kne, the class of all orgin-symmetric convex bodies (cf. Section 2), there exists a unique convex body ˉK∈Kne such that
∫Sn−1g(u)dS(K,u)=∫Sn−1g(u)dSp(ˉK,u) | (1.11) |
for all continuous functions g on Sn−1. Here Sp(ˉK,⋅) is Lp surface area measure of ˉK. In fact, it is the obvious result by applying the even Lp Minkowski problem (cf. Section 4). Moreover, Ψ:Kne→Kne, Ψ(K)=ˉK, is a bijection. On the other hand, the Lp projection body is introduced, which is the analogue of the projection body, for convex bodies. The Lp Petty projection inequality is used, which is the analogue of the Petty projection inequality, for convex bodies (cf. Section 5).
It follows from the H¨older inequality and the Cauchy surface area formula that the affine Sobolev-Zhang inequality implies the sharp Sobolev inequality. Similarly, the inequality (1.7) is stronger than the sharp Sobolev inequality on BV(Rn) by the H¨older inequality and the Lp Cauchy surface area formula (cf. Section 5).
Our setting will be n-dimensional Euclidean space Rn where n≥2. The standard inner product of the vectors x,y∈Rn is denoted by x⋅y. The standard Euclidean norm of the vector x∈Rn is denoted by |x|. The closed ball with center x and radius r is denoted by Br(x), and we write Sn−1 for the topological boundary of the Euclidean unit ball B1(0). Let k be a non-negative integer. By Hk we denote the k-dimensional Hausdorff measure on Rn, and Hn is equal to the n-dimensional Lebesgue measure. The scalar multiple of a set A⊂Rn is defined by
λA={λa:a∈A} |
for real numbers λ. We write −A for (−1)A.
In this section, we collect some notations and basic facts about convex bodies (see, e.g., [12,13,32]).
A convex body is a compact convex subset of Rn with non-empty interior. The class of convex bodies is denoted by Kn. The class of convex bodies containing the origin in their interiors is denoted by Kno. The set K is called origin-symmetric if K=−K. Let Kne denote the class of origin-symmetric convex bodies in Rn. The set K is called symmetric if some translation of K is origin-symmetric. Each non-empty compact convex set K is uniquely determined by its support function hK, defined by
hK(x)=sup{x⋅y:y∈K} |
for all x∈Rn. Note that hK is positively homogeneous of degree 1 and subadditive. Conversely, each function with these two properties is the support function of a unique compact convex set. For K∈Kne, it is obviously ture that the support function of K is even, that is, hK(x)=hK(−x) for all x∈Rn.
Let K be a convex body and ν:∂K→Sn−1 the generalized Gauss map (ν is set-valued), where ∂K is the topological boundary of K. Note that ν(x) is the set of all outer unit normal vectors at boundary point x of K. For each Borel set ω⊂Sn−1, the inverse spherical image ν−1(ω) of ω is the set of all boundary points of K which have an outer unit normal vector belonging to the set ω. It is easy to check that ν−1(ω) is measurable. Associated with each convex body K a Borel measure S(K,⋅) on Sn−1 called the surface area measure of K, is defined by
S(K,ω)=Hn−1(ν−1(ω)), |
for each Borel set ω⊂Sn−1.
The mixed volume V1(K,L) of two convex bodies K and L is defined by
V1(K,L)=1n∫Sn−1hL(u)dS(K,u). | (2.1) |
In this paper, V and S stand for the volume functional and the surface area functional on Kn, respectively. It is easy to see both
V1(K,K)=V(K)=Hn(K) |
and
nV1(K,B1(0))=S(K)=Hn−1(∂K). |
A fundamental inequality which will be used is the first Minkowski inequality.
Lemma 2.1. ([13,p. 101])If K,L∈Kn, then
V1(K,L)≥V(K)n−1nV(L)1n, | (2.2) |
where equality holds if and only if K and L are positive homothetic.
Here K and L are positive homothetic, if there exist λ>0 and x∈Rn such that
L=λK+x={λy+x:y∈K}. |
For K∈Kno, the polar body K∘ of K is defined by
K∘={x∈Rn:x⋅y≤1 for all y∈K}. |
A formula of the volume of the polar body K∘ is that
V(K∘)=1n∫Sn−1h−nKdu. | (2.3) |
The projection body of K∈Kn is the convex body whose support function is defined by
hΠK(x)=12∫Sn−1|u⋅x|dS(K,u), x∈Rn. |
Note that
hΠK(v)=Hn−1(K|v⊥), |
where v∈Sn−1, K|v⊥ is orthogonal projection of K onto the linear subspace orthogonal to v and Hn−1(K|v⊥) is the volume of K|v⊥ in the (n−1)-dimensional linear subspace. For convenience, (ΠK)∘ is denoted by Π∘K.
Next, some notations of the Lp Brunn-Minkowski theory are introduced.
Given K∈Kno and p∈R, a Borel measure Sp(K,⋅) on Sn−1 called the Lp surface area measure of K, is defined by
Sp(K,ω)=h1−pKS(K,ω) |
for each Borel set ω⊂Sn−1. Obviously, S1(K,⋅)=S(K,⋅). We denote the total measure Sp(K,Sn−1) of the Lp surface area measure of K by Sp(K).
For p≥1, the Lp mixed volume Vp(K,L) of two convex bodies K,L∈Kno is defined by
Vp(K,L)=1n∫Sn−1hpL(u)dSp(K,u). | (2.4) |
Note that, particularly, Vp(K,K)=V(K).
For p≥1, the Lp projection body of K∈Kno is the convex body whose support function is defined by
hpΠpK(x)=1cn−2,pnωn∫Sn−1|u⋅x|pdSp(K,u), x∈Rn. |
Here cn−2,p is consistent with (1.8). The normalization of ΠpK is such that ΠpB1(0)=B1(0); therefore, Π1=ω−1n−1Π≠Π. Similarly, we write Π∘pK for (ΠpK)∘.
In this section, we review some basic notations and facts about functions of bounded variation on Rn (see [9]).
Throughout this paper, C1c(Rn;Rn) stands for the class of the compactly supported continuously differentiable functions from Rn to Rn and Lp(Rn) contains all Lebesgue measurable functions f with
‖f‖p=(∫Rn|f(x)|pdx)1p<∞. |
Definition 3.1. A function f∈L1(Rn) has bounded variation if
sup{∫Rnfdivϕdx:ϕ∈C1c(Rn;Rn),|ϕ|≤1}<∞. |
We write
BV(Rn) |
to denote the space of functions of bounded variation.
Here div denotes the divergence operator.
Theorem 3.2. ([9,p. 194,Structure Theorem for BV functions])Assume that f∈BV(Rn). Then there exist a Radon measure μ on Rn anda μ-measurable function
σf:Rn→Rn |
such that
(1) |σf(x)|=1 μ-a.e., and
(2) for all ϕ∈C1c(Rn;Rn), we have
∫Rnfdivϕdx=−∫Rnϕ⋅σfdμ. |
Hence, we will write |Df| for the measure μ, and Df:=σf|Df|. While
σf(x)=limr→0+Df(Br(x))|Df|(Br(x)) |
for x∈Rn a.e., with respect to |Df|.
For example, each Sobolev function on Rn has bounded variation, that is,
W1,1(Rn)⊂BV(Rn). |
If μ is the surface area measure S(K,⋅) of the convex body K∈Kn, then
∫Sn−1udμ(u)=o, | (4.1) |
where o is the origin. And it is clear that
μ cannot be concentrated on any great subsphere of Sn−1. | (4.2) |
Conversely, the Minkowski problem was proposed, which reads as follows:
Find necessary and sufficient conditions on a finite Borel measure μ on the unit sphere Sn−1 so that μ is the surface area measure of a convex body K∈Kn.
Fortunately, (4.1) and (4.2) are also sufficient in order that μ be the surface area measure of a convex body K∈Kn. Moreover, we have uniqueness, that is, the K is unique up to translations.
In recent years, for various Minkowski problems, a large number of related results have been obtained (see, e.g., [18,19,28,35,36,39]).
In [33], the author introduced the extended LYZ body ⟨f⟩ of f∈BV(Rn) by using the solution of the classical Minkowski problem.
Definition 4.1. ([33]) For f∈BV(Rn) which is not 0, the LYZ body is defined to be the origin-symmetric convex body ⟨f⟩, such that
∫Sn−1g(u)dS(⟨f⟩,u)=∫Rng(σf)d|Df| | (4.3) |
for every g:Rn→R that is even, continuous and positively 1-homogeneous.
We can write a convex body K using its characteristic function 1K. And the characteristic function 1K has bounded variation. Then, we have Kn⊂BV(Rn) in this sense. Hence, the LYZ operator Φ:BV(Rn)→Kne, Φ(f)=⟨f⟩ can be regarded as a operator from BV(Rn) to BV(Rn).
Now, we collect some properties of the LYZ operator on BV(Rn).
Lemma 4.2. ([33])For t∈R+ and f∈BV(Rn), we have ⟨tf⟩=t1n−1⟨f⟩, ⟨−f⟩=⟨f⟩.
Lemma 4.3. ([33])Given K∈Kne, if K+x is the translation of K with respect to x∈Rn, we have ⟨1K+x⟩=K.
Lemma 4.4. ([33])Let f∈BV(Rn). Then
V(⟨f⟩)n−1n≥‖f‖nn−1, | (4.4) |
where there is equality if and only if f is a multiple of the characteristic function of a symmetric convex body.
We can rewrite (4.4) as
‖1⟨f⟩‖nn−1≥‖f‖nn−1. |
Thus, Lemma 4.4 guarantees that the Lnn−1 norm of f∈BV(Rn) is increased by the LYZ operator, while the LYZ operator keeps
∫Rng(σf)d|Df|=∫Rng(σ1⟨f⟩)d|D1⟨f⟩| | (4.5) |
for every g:Rn→R that is even, continuous and positively 1-homogeneous.
Next, we introduce the Lp Minkowski problem. Using the solution of the even Lp Minkowski problem, we can get a bijective operator Ψ:Kne→Kne (see Theorem 4.7).
Analogously, the Lp Minkowski problem is that for p∈R find necessary and sufficient conditions on a finite Borel measure μ on the unit sphere Sn−1 so that μ is the Lp surface area measure of a convex body K∈Kn.
Particularly, the even Lp Minkowski problem is of great interest, which reads as follows:
Find necessary and sufficient conditions on an even finite Borel measure μ on the unit sphere Sn−1 so that μ is the even Lp surface area measure of an origin-symmetric convex body K.
Here, the measure μ is even if μ(−ω)=μ(ω) for each Borel set ω⊂Sn−1. For p>0, the problem has been solved. We have the following result.
Lemma 4.5. ([32,p. 498])Let p>0 and p≠n. Let μ be an even finite Borel measure on Sn−1 which cannot be concentrated on any great subsphere of Sn−1. Then, there exists K∈Kne such that Sp(K,⋅)=μ.
Moreover, for suitable p, the solution of the even Lp Minkowski problem is unique, which holds true by
Lemma 4.6. ([32,p. 494])Let p>1, p≠n, and K,L∈Kno. If
Sp(K,⋅)=Sp(L,⋅), |
then K=L.
Using the solution of the even Lp Minkowski problem, we associate each orgin-symmetric convex body K with the another orgin-symmetric convex body, which reads as follows:
Theorem 4.7. Given p>1, p≠n and K∈Kne, there exists a unique convex body ˉK∈Kne such that
∫Sn−1g(u)dS(K,u)=∫Sn−1g(u)dSp(ˉK,u) | (4.6) |
for all continuous functions g on Sn−1. Moreover, the operator Ψ:Kne→Kne, Ψ(K)=ˉK is a bijection.
Proof. Let K∈Kne, p>1 and p≠n. The surface area measure S(K,⋅) of K satisfies the necessary and sufficient conditions of the even Lp Minkowski problem. Using Lemmas 4.5 and 4.6, then there exists a unique origin-symmetric convex body ˉK such that
S(K,⋅)=Sp(ˉK,⋅). |
Thus, we have
∫Sn−1g(u)dS(K,u)=∫Sn−1g(u)dSp(ˉK,u) |
for all continuous functions g on Sn−1.
If K,K1∈Kne and Ψ(K)=Ψ(K1)=ˉK, then we have
S(K,⋅)=Sp(ˉK,⋅)=S(K1,⋅). |
By the uniqueness the solution of the Minkowski problem, K=K1 up to some translation. Since K and K1 are origin-symmetric, K=K1. Thus Ψ is bijective.
Now, we deduce some properties of the operator Ψ.
It follows from the homogeneity of Hausdorff measures and the homogeneity of support functions that Sp(λK,⋅)=λn−pSp(K,⋅) for p≥1, λ>0 and K∈Kno.
The following lemma states that Ψ(B) is also a ball with center at o for a ball B∈Kne.
Lemma 4.8. Let K∈Kne. If K=rB1(0) for some r>0, then Ψ(K)=ˉK=rn−1n−pB1(0).
Proof. By the definition and the homogeneity of the surface area measure, we have
∫Sn−1g(u)dS(rB1(0),u)=∫Sn−1g(u)rn−1dS(B1(0),u) |
for all continuous functions g on Sn−1. Since hB1(0)=1, Sp(L,⋅)=h1−pLS(L,⋅) for some L∈Kno and Sp(λK,⋅)=λn−pSp(K,⋅), we have
∫Sn−1g(u)rn−1dS(B1(0),u)=∫Sn−1g(u)rn−1dSp(B1(0),u)=∫Sn−1g(u)(rn−1n−p)n−pdSp(B1(0),u)=∫Sn−1g(u)dSp(rn−1n−pB1(0),u). |
Then, we have
∫Sn−1g(u)dS(rB1(0),u)=∫Sn−1g(u)dSp(rn−1n−pB1(0),u) |
for all continuous functions g on Sn−1. That is Ψ(rB1(0))=rn−1n−pB1(0).
Since the LYZ operator increases the Lnn−1 norm of a function which has bounded variation, similarly, the Ψ operator changes the volume of a origin-symmetric convex body. We have the following inequality.
Theorem 4.9. Let K∈Kne, p>1, p≠n and 1p+1q=1. Then
V(ˉK)n−pnp≥V(K)n−1n(S(K)n)−1q, | (4.7) |
where equality holds if and only if K is a Euclidean ball with center at o.
Remark 4.10. Note that S(K)=Sp(ˉK). The inequality (4.7) can be rewritten as
(nV(ˉK)Sp(ˉK))n−pnp≥(nV(K)S(K))n−1n. |
Proof. Let K,L∈Kne, p>1 and p≠n. By the H¨older inequality, we have
∫Sn−1hL(u)dS(K,u)≤(∫Sn−1hpL(u)dS(K,u))1/pS(K)1/q, | (4.8) |
where equality holds if and only if L=Br(0) for some r>0. Since
∫Sn−1hpL(u)dS(K,u)=∫Sn−1hpL(u)dSp(ˉK,u), |
by Theorem 4.7, we have
∫Sn−1hL(u)dS(K,u)≤(∫Sn−1hpL(u)dSp(ˉK,u))1/pS(K)1/q. | (4.9) |
Setting L=ˉK in the inequality (4.9), we get
nV1(K,ˉK)≤(nV(ˉK))1/pS(K)1/q. | (4.10) |
It follows from the equality condition in (4.8) and Lemma 4.8 that the equality holds in (4.10) if and only if K=Br(0) for some r>0. Using Lemma 2.1, we have
(nV(ˉK))1pS(K)1q≥nV1(K,ˉK)≥nV(K)n−1nV(ˉK)1n. | (4.11) |
Then
V(ˉK)n−pnp≥V(K)n−1n(S(K)n)−1q. |
If K=rB1(0) for some r>0, then ˉK=rn−1n−pB1(0). By the equality condition in (4.10) and the equality condition in Lemma 2.1, we have
V(ˉK)n−pnp=V(K)n−1n(S(K)n)−1q. |
If K≠rB1(0) for each r>0, the equality does not hold in the inequality (4.10). Then
V(ˉK)n−pnp>V(K)n−1n(S(K)n)−1q. |
Thus, we have
V(ˉK)n−pnp=V(K)n−1n(S(K)n)−1q, |
if and only if K is a ball with center at o.
As a consequence of Lemma 4.4 and Theorem 4.9, we obtain:
Corollary 4.11. Let f∈BV(Rn), f≠0, p>1, p≠n and 1p+1q=1. Then
‖f‖nn−1(‖Df‖n)−1q≤V(¯⟨f⟩)n−pnp, |
where equality holds if and only if f=λ1B for some λ∈Rn and some Euclidean ball B.
Proof.
Let f∈BV(Rn), f≠0, p>1, p≠n and 1p+1q=1. Using Theorem 4.9 and Lemma 4.4, we have
V(¯⟨f⟩)n−pnp≥V(⟨f⟩)n−1n(S(⟨f⟩)n)−1q≥‖f‖nn−1(S(⟨f⟩)n)−1q. |
It follows from ‖Df‖=S(⟨f⟩) that
‖f‖nn−1(‖Df‖n)−1q≤V(¯⟨f⟩)n−pnp. |
Let f=λ1B for some λ∈Rn and some Euclidean ball B. Using the equality conditions in Theorem 4.9 and Lemma 4.4, we get
‖f‖nn−1(‖Df‖n)−1q=V(¯⟨f⟩)n−pnp. |
Now, assume that
‖f‖nn−1(‖Df‖n)−1q=V(¯⟨f⟩)n−pnp |
for f∈BV(Rn) and f≠0. Thus
‖f‖nn−1(‖Df‖n)−1q=V(⟨f⟩)n−1n(S(⟨f⟩)n)−1q=V(¯⟨f⟩)n−pnp. |
By Theorem 4.9 and Lemma 4.4, we have ⟨f⟩=rB1(0) for some r>0 and f=λ1K+x for some K∈Kne and some x∈Rn. Using Lemmas 4.2 and 4.3, we get
rB1(0)=⟨f⟩=⟨λ1K+x⟩=λ1n−1⟨1K+x⟩=λ1n−1K. |
Thus, K=(r/λ1n−1)B1(0). Then, we get f=λ1B for the ball B=(r/λ1n−1)B1(0)+x.
Remark 4.12. Although the Lnn−1 norm of f∈BV(Rn) is variant under the operator LYZ and the operator Ψ, the variation measure |Df| satisfies some invariance. We have
∫Rng(σf)d|Df|=∫Sn−1g(u)dS(⟨f⟩,u)=∫Sn−1g(u)dSp(¯⟨f⟩,u), | (4.12) |
for every g:Rn→R that is even, continuous and positively 1-homogeneous.
In this section, we prove Theorem 1.1 and show that the inequality (1.7) implies the Sobolev inequality on BV(Rn).
We will use the Lp Petty projection inequality, which is proved by Lutwak-Yang-Zhang in [26]. It is the Lp analogue of the Petty projection inequality.
Theorem 5.1. ([26] or [32,p. 575]) For 1<p<∞ and for K∈Kno,
V(K)(n−p)/pV(Π∘pK)≤ωn/pn, |
with equality if and only if K is an origin-symmetric ellipsoid.
Now, we prove Theorem 1.1.
Proof. Let f∈BV(Rn), f≠0, p>1, p≠n and 1p+1q=1. By (2.3) and Remark 4.12, we calculate
V(Π∘p¯⟨f⟩)=1n∫Sn−1h−nΠp¯⟨f⟩du=1n∫Sn−1(1cn−2,pnωn∫Sn−1|u⋅v|pdSp(¯⟨f⟩,v))−npdu=1n∫Sn−1(1cn−2,pnωn∫Rn|u⋅σf|pd|Df|)−npdu=(cn−2,pnωn)np1n∫Sn−1(∫Rn|u⋅σf|pd|Df|)−npdu. | (5.1) |
Using Corollary 4.11, we have
‖f‖nnn−1(‖Df‖n)−nq≤V(¯⟨f⟩)n−pp, | (5.2) |
where equality holds if and only if f=λ1B for some λ∈Rn and some Euclidean ball B. Thus, it follows from (5.2) and (5.1) that
∫Sn−1(∫Rn|u⋅σf|pd|Df|)−npdu≤V(¯⟨f⟩)n−ppV(Π∘p¯⟨f⟩)‖f‖−nnn−1‖Df‖nq(1n)n−1(1cn−2,pωn)np, |
i.e.,
∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu≤V(¯⟨f⟩)n−ppV(Π∘p¯⟨f⟩)ωn/pn(‖Df‖‖f‖1∗)n(1n)n−1(1cn−2,p)np, | (5.3) |
where 1∗=n/(n−1) and equality holds if and only if f=λ1B for some λ∈Rn and some Euclidean ball B. Applying Theorem 5.1, we see
V(¯⟨f⟩)n−ppV(Π∘p¯⟨f⟩)ωn/pn≤1, | (5.4) |
with equality if and only if ¯⟨f⟩ is an origin-symmetric ellipsoid. Then, it directly follows from (5.4) and (5.3) that
∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu≤(1n)n−1(1cn−2,p)np(‖Df‖‖f‖1∗)n. | (5.5) |
If f=λ1B for some λ∈Rn and some Euclidean ball B, then ¯⟨f⟩=λ1n−prn−1n−pB1(0), where r is the radius of B. Hence, the equality holds in (5.4), so the equality holds in (5.5).
If the equality holds in (5.5), then the equality holds in (5.3) and (5.2). So f=λ1B for some λ∈Rn and some Euclidean ball B.
In summary, the equality holds in (5.5) if and only if f=λ1B for some λ∈Rn and some Euclidean ball B.
Now, let p→n in the inequality (5.5). It follows from the dominated convergence theorem that
∫Sn−1( −∫Rn|u⋅σf|nd|Df|)−1du≤(1n)n−1(1cn−2,n)(‖Df‖‖f‖1∗)n. | (5.6) |
Since
∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu=(1n)n−1(1cn−2,p)np(‖Df‖‖f‖1∗)n |
with p≠n and f=λ1B for some λ∈Rn and some Euclidean ball B, we have
∫Sn−1( −∫Rn|u⋅σf|nd|Df|)−1du=(1n)n−1(1cn−2,n)(‖Df‖‖f‖1∗)n |
by p→n for f=λ1B.
Now, we prove that the inequality (1.7) or (5.5) is stronger than the sharp Sobolev inequality on BV(Rn).
The sharp Sobolev inequality on BV(Rn) states that
Corollary 5.2. Let f∈BV(Rn). Then
nω1/nn‖f‖1∗≤‖Df‖, | (5.7) |
where equality holds if and only if f=λ1B for some λ∈Rn and some Euclidean ball B⊂Rn.
Firstly, we deduce the Lp Cauchy surface area formula, which is a direct consequence by the following lemma.
Lemma 5.3. Let K,L∈Kno and p≥1. Then
Vp(L,ΠpK)=Vp(K,ΠpL). | (5.8) |
Proof. Using (2.4) and Fubini's theorem, we directly calculate
Vp(L,ΠpK)=1n∫Sn−1hpΠpK(u)dSp(L,u)=1n∫Sn−11cn−2,pnωn∫Sn−1|u⋅v|pdSp(K,v)dSp(L,u)=1n∫Sn−11cn−2,pnωn∫Sn−1|u⋅v|pdSp(L,u)dSp(K,v)=1n∫Sn−1hpΠpL(v)dSp(K,v)=Vp(K,ΠpL). |
The Lp Cauchy surface area formula is the following.
Theorem 5.4. Let p≥1 and K∈Kno. Then
Sp(K)=1cn−2,pnωn∫Sn−1(∫Sn−1|u⋅v|pdSp(K,v))du. |
Proof. Let p≥1 and K∈Kno. Setting L=B1(0) in the equality (5.8), we have
Vp(B1(0),ΠpK)=Vp(K,ΠpB1(0)). |
By ΠpB1(0)=B1(0), then
Vp(B1(0),ΠpK)=Vp(K,B1(0)). |
Thus, we can compute
Sp(K)=nVp(K,B1(0))=nVp(B1(0),ΠpK)=∫Sn−1hpΠpK(u)dSp(B1(0),u)=∫Sn−1hpΠpK(u)du=1cn−2,pnωn∫Sn−1(∫Sn−1|u⋅v|pdSp(K,v))du. |
Remark 5.5. Let p=1 in Theorem 5.4 and note that 2ω2ωn−2=nωn. We get the classical Cauchy surface area formula, which reads as follows: If K∈Kn, then
S(K)=1ωn−1∫Sn−1Hn−1(K|u⊥)du. |
Now, we prove Corollary 5.2 by using Theorem 1.1 for fixed p>1 with p≠n.
Proof. Let f∈BV(Rn). If f=0, then the inequality (1.1) holds trivially.
Assume f≠0 and p>1. From Remark 4.12 and Theorem 5.4, we have
cn−2,pnωn=∫Sn−1(1Sp(¯⟨f⟩)∫Sn−1|u⋅v|pdSp(¯⟨f⟩,v))du=∫Sn−1(1‖Df‖∫Rn|u⋅σf|pd|Df|)du=∫Sn−1( −∫Rn|u⋅σf|pd|Df|)du. | (5.9) |
We set β=nn+p, that is, −pn+1β=1. It follows from the H¨older inequality that
∫Sn−1( −∫Rn|u⋅σf|pd|Df|)du≥(∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu)−pn(nωn)(n+p)/n, |
where equality holds if and only if
−∫Rn|u⋅σf|pd|Df|=constant, |
that is, Πp¯⟨f⟩ is a Euclidean ball with center at o. Thus, combining with (5.9), we see that
∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu≥(1cn−2,p)npnωn. |
Now, using Theorem 1.1, we calculate
(1n)n−1(1cn−2,p)np(‖Df‖‖f‖1∗)n≥∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu≥(1cn−2,p)npnωn, |
that is,
‖Df‖≥nω1/nn‖f‖1∗. |
If f=λ1B for some λ∈Rn and some Euclidean ball B⊂Rn, then the equality holds in the inequality (1.7) and Πp¯⟨f⟩ is a Euclidean ball with center at o. Thus,
(1n)n−1(1cn−2,p)np(‖Df‖‖f‖1∗)n=∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu=(1cn−2,p)npnωn, |
i.e.,
‖Df‖=nω1/nn‖f‖1∗. |
If p≠n and f≠λ1B for all λ∈Rn and all Euclidean balls B⊂Rn, then
(1n)n−1(1cn−2,p)np(‖Df‖‖f‖1∗)n>∫Sn−1( −∫Rn|u⋅σf|pd|Df|)−npdu. |
Thus,
‖Df‖>nω1/nn‖f‖1∗. |
In summary, for p≠n we have that
‖Df‖=nω1/nn‖f‖1∗ |
if and only if f=λ1B for some λ∈Rn and some Euclidean ball B⊂Rn.
In this work, we establish a family of new Sobolev inequalities on BV(Rn), and we prove that each one in the family can imply the classical Sobolev inequality with the sharp constant on BV(Rn) which is one of the most important inequality in analysis. Our approach is the Lp Brunn-Minkowski theory in convex geometry. We use the Lutwak-Yang-Zhang operator so that inequalities of BV functions relate to inequalities of convex bodies. Then, we establish a family of inequalities of convex bodies. As a consequence, we achieve the goal.
The work of the authors was supported by the Recruitment Program for Young Professionals of China and the Fundamental Research Funds for the Central Universities (Grant No. GK202101008) and Postgraduate Innovation Team Project of Shaanxi Normal University (No. TD2020008Z).
The authors declare that there is no conflict of interests regarding the publication of this article.
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