Research article

Some sharp Sobolev inequalities on BV(Rn)

  • Received: 09 May 2022 Revised: 28 June 2022 Accepted: 04 July 2022 Published: 14 July 2022
  • MSC : 46B20, 46E35, 52A21

  • In this paper, some sharp Sobolev inequalities on BV(Rn), the space of functions of bounded variation on Rn, n2, 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, n2, 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), n2, going back to [10,11,30], states that

    nω1/nnf1|f|1,

    where 1=n/(n1), |f| is the Euclidean norm of the weak gradient of f, fp is the usual Lp norm of f in Rn and ωk is the volume enclosed by the unit sphere Sk1 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 fBV(Rn), then we have

    nω1/nnf1Df, (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:

    Sn1(Rn|uσf|pd|Df|)npdu(1n)n1(1cn2,p)npfn1Dfn(p1)p, (1.2)

    for fBV(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, n2, which states

    1nSn1ufn1du(ωn2ωn1)nfn1. (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 fBV(Rn),

    1nSn1(Rn|uσf|d|Df|)ndu(ωn2ωn1)nfn1. (1.4)

    It is proved that equality holds if and only if f=λ1E for some λR and some ellipsoid ERn. Here 1A denotes the characteristic function of ARn.

    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)=(1nSn1ufnpdu)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 ERn is a μ-measurable set with μ(E)<, then we set

    Efdμ=1μ(E)Efdμ.

    Now assuming f0, we can rewrite the inequality (1.4) as

    1nSn1( Rn|uσf|d|Df|)ndu(ωn2ωn1)n(Dff1)n. (1.5)

    For p>1, using the H¨older inequality, we have

    Rn|uσf|d|Df|( Rn|uσf|pd|Df|)1p,

    where equality holds if and only if |uσf|=constant. Then (1.5) implies

    1nSn1( Rn|uσf|pd|Df|)npdu(ωn2ωn1)n(Dff1)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 λR and any ellipsoid E, while |uσλ1E| is never equal to some constant indentically.

    So, in this paper, we refine the constant (ωn/(2ωn1))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 fBV(Rn), f0 and p>1. Then

    Sn1( Rn|uσf|pd|Df|)npdu(1n)n1(1cn2,p)np(Dff1)n, (1.7)

    where equality holds whenever f=λ1B for some λR and some ball BRn. Moreover, for pn, if the equality holds in (1.7), then f=λ1B for some λR and some ball BRn.

    Here

    cn2,p=ωn+p2ω2ωn2ωp1, (1.8)

    ωs=πs/2/Γ(1+s/2) for s0 and Γ() is the Gamma function. Note that if p1+ 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 fW1,1(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 fW1,1(Rn), there exists a unique origin-symmetric convex body f such that

    Sn1g(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 fBV(Rn), there exists a unique origin-symmetric convex body f such that

    Sn1g(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, pn and KKne, the class of all orgin-symmetric convex bodies (cf. Section 2), there exists a unique convex body ˉKKne such that

    Sn1g(u)dS(K,u)=Sn1g(u)dSp(ˉK,u) (1.11)

    for all continuous functions g on Sn1. 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:KneKne, 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 n2. The standard inner product of the vectors x,yRn is denoted by xy. The standard Euclidean norm of the vector xRn is denoted by |x|. The closed ball with center x and radius r is denoted by Br(x), and we write Sn1 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 ARn is defined by

    λA={λa:aA}

    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{xy:yK}

    for all xRn. 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 KKne, it is obviously ture that the support function of K is even, that is, hK(x)=hK(x) for all xRn.

    Let K be a convex body and ν:KSn1 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 ωSn1, the inverse spherical image ν1(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 ν1(ω) is measurable. Associated with each convex body K a Borel measure S(K,) on Sn1 called the surface area measure of K, is defined by

    S(K,ω)=Hn1(ν1(ω)),

    for each Borel set ωSn1.

    The mixed volume V1(K,L) of two convex bodies K and L is defined by

    V1(K,L)=1nSn1hL(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)=Hn1(K).

    A fundamental inequality which will be used is the first Minkowski inequality.

    Lemma 2.1. ([13,p. 101])If K,LKn, then

    V1(K,L)V(K)n1nV(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 xRn such that

    L=λK+x={λy+x:yK}.

    For KKno, the polar body K of K is defined by

    K={xRn:xy1 for all yK}.

    A formula of the volume of the polar body K is that

    V(K)=1nSn1hnKdu. (2.3)

    The projection body of KKn is the convex body whose support function is defined by

    hΠK(x)=12Sn1|ux|dS(K,u),   xRn.

    Note that

    hΠK(v)=Hn1(K|v),

    where vSn1, K|v is orthogonal projection of K onto the linear subspace orthogonal to v and Hn1(K|v) is the volume of K|v in the (n1)-dimensional linear subspace. For convenience, (ΠK) is denoted by ΠK.

    Next, some notations of the Lp Brunn-Minkowski theory are introduced.

    Given KKno and pR, a Borel measure Sp(K,) on Sn1 called the Lp surface area measure of K, is defined by

    Sp(K,ω)=h1pKS(K,ω)

    for each Borel set ωSn1. Obviously, S1(K,)=S(K,). We denote the total measure Sp(K,Sn1) of the Lp surface area measure of K by Sp(K).

    For p1, the Lp mixed volume Vp(K,L) of two convex bodies K,LKno is defined by

    Vp(K,L)=1nSn1hpL(u)dSp(K,u). (2.4)

    Note that, particularly, Vp(K,K)=V(K).

    For p1, the Lp projection body of KKno is the convex body whose support function is defined by

    hpΠpK(x)=1cn2,pnωnSn1|ux|pdSp(K,u),   xRn.

    Here cn2,p is consistent with (1.8). The normalization of ΠpK is such that ΠpB1(0)=B1(0); therefore, Π1=ω1n1ΠΠ. Similarly, we write ΠpK 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

    fp=(Rn|f(x)|pdx)1p<.

    Definition 3.1. A function fL1(Rn) has bounded variation if

    sup{Rnfdivϕdx:ϕC1c(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 fBV(Rn). Then there exist a Radon measure μ on Rn anda μ-measurable function

    σf:RnRn

    such that

    (1) |σf(x)|=1 μ-a.e., and

    (2) for all ϕC1c(Rn;Rn), we have

    Rnfdivϕdx=Rnϕσfdμ.

    Hence, we will write |Df| for the measure μ, and Df:=σf|Df|. While

    σf(x)=limr0+Df(Br(x))|Df|(Br(x))

    for xRn 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 KKn, then

    Sn1udμ(u)=o, (4.1)

    where o is the origin. And it is clear that

    μ cannot be concentrated on any great subsphere of Sn1. (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 Sn1 so that μ is the surface area measure of a convex body KKn.

    Fortunately, (4.1) and (4.2) are also sufficient in order that μ be the surface area measure of a convex body KKn. 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 fBV(Rn) by using the solution of the classical Minkowski problem.

    Definition 4.1. ([33]) For fBV(Rn) which is not 0, the LYZ body is defined to be the origin-symmetric convex body f, such that

    Sn1g(u)dS(f,u)=Rng(σf)d|Df| (4.3)

    for every g:RnR 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 KnBV(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 tR+ and fBV(Rn), we have tf=t1n1f, f=f.

    Lemma 4.3. ([33])Given KKne, if K+x is the translation of K with respect to xRn, we have 1K+x=K.

    Lemma 4.4. ([33])Let fBV(Rn). Then

    V(f)n1nfnn1, (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

    1fnn1fnn1.

    Thus, Lemma 4.4 guarantees that the Lnn1 norm of fBV(Rn) is increased by the LYZ operator, while the LYZ operator keeps

    Rng(σf)d|Df|=Rng(σ1f)d|D1f| (4.5)

    for every g:RnR 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:KneKne (see Theorem 4.7).

    Analogously, the Lp Minkowski problem is that for pR find necessary and sufficient conditions on a finite Borel measure μ on the unit sphere Sn1 so that μ is the Lp surface area measure of a convex body KKn.

    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 Sn1 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 ωSn1. For p>0, the problem has been solved. We have the following result.

    Lemma 4.5. ([32,p. 498])Let p>0 and pn. Let μ be an even finite Borel measure on Sn1 which cannot be concentrated on any great subsphere of Sn1. Then, there exists KKne 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, pn, and K,LKno. 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, pn and KKne, there exists a unique convex body ˉKKne such that

    Sn1g(u)dS(K,u)=Sn1g(u)dSp(ˉK,u) (4.6)

    for all continuous functions g on Sn1. Moreover, the operator Ψ:KneKne, Ψ(K)=ˉK is a bijection.

    Proof. Let KKne, p>1 and pn. 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

    Sn1g(u)dS(K,u)=Sn1g(u)dSp(ˉK,u)

    for all continuous functions g on Sn1.

    If K,K1Kne 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,)=λnpSp(K,) for p1, λ>0 and KKno.

    The following lemma states that Ψ(B) is also a ball with center at o for a ball BKne.

    Lemma 4.8. Let KKne. If K=rB1(0) for some r>0, then Ψ(K)=ˉK=rn1npB1(0).

    Proof. By the definition and the homogeneity of the surface area measure, we have

    Sn1g(u)dS(rB1(0),u)=Sn1g(u)rn1dS(B1(0),u)

    for all continuous functions g on Sn1. Since hB1(0)=1, Sp(L,)=h1pLS(L,) for some LKno and Sp(λK,)=λnpSp(K,), we have

    Sn1g(u)rn1dS(B1(0),u)=Sn1g(u)rn1dSp(B1(0),u)=Sn1g(u)(rn1np)npdSp(B1(0),u)=Sn1g(u)dSp(rn1npB1(0),u).

    Then, we have

    Sn1g(u)dS(rB1(0),u)=Sn1g(u)dSp(rn1npB1(0),u)

    for all continuous functions g on Sn1. That is Ψ(rB1(0))=rn1npB1(0).

    Since the LYZ operator increases the Lnn1 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 KKne, p>1, pn and 1p+1q=1. Then

    V(ˉK)npnpV(K)n1n(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))npnp(nV(K)S(K))n1n.

    Proof. Let K,LKne, p>1 and pn. By the H¨older inequality, we have

    Sn1hL(u)dS(K,u)(Sn1hpL(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

    Sn1hpL(u)dS(K,u)=Sn1hpL(u)dSp(ˉK,u),

    by Theorem 4.7, we have

    Sn1hL(u)dS(K,u)(Sn1hpL(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)1qnV1(K,ˉK)nV(K)n1nV(ˉK)1n. (4.11)

    Then

    V(ˉK)npnpV(K)n1n(S(K)n)1q.

    If K=rB1(0) for some r>0, then ˉK=rn1npB1(0). By the equality condition in (4.10) and the equality condition in Lemma 2.1, we have

    V(ˉK)npnp=V(K)n1n(S(K)n)1q.

    If KrB1(0) for each r>0, the equality does not hold in the inequality (4.10). Then

    V(ˉK)npnp>V(K)n1n(S(K)n)1q.

    Thus, we have

    V(ˉK)npnp=V(K)n1n(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 fBV(Rn), f0, p>1, pn and 1p+1q=1. Then

    fnn1(Dfn)1qV(¯f)npnp,

    where equality holds if and only if f=λ1B for some λRn and some Euclidean ball B.

    Proof.

    Let fBV(Rn), f0, p>1, pn and 1p+1q=1. Using Theorem 4.9 and Lemma 4.4, we have

    V(¯f)npnpV(f)n1n(S(f)n)1qfnn1(S(f)n)1q.

    It follows from Df=S(f) that

    fnn1(Dfn)1qV(¯f)npnp.

    Let f=λ1B for some λRn and some Euclidean ball B. Using the equality conditions in Theorem 4.9 and Lemma 4.4, we get

    fnn1(Dfn)1q=V(¯f)npnp.

    Now, assume that

    fnn1(Dfn)1q=V(¯f)npnp

    for fBV(Rn) and f0. Thus

    fnn1(Dfn)1q=V(f)n1n(S(f)n)1q=V(¯f)npnp.

    By Theorem 4.9 and Lemma 4.4, we have f=rB1(0) for some r>0 and f=λ1K+x for some KKne and some xRn. Using Lemmas 4.2 and 4.3, we get

    rB1(0)=f=λ1K+x=λ1n11K+x=λ1n1K.

    Thus, K=(r/λ1n1)B1(0). Then, we get f=λ1B for the ball B=(r/λ1n1)B1(0)+x.

    Remark 4.12. Although the Lnn1 norm of fBV(Rn) is variant under the operator LYZ and the operator Ψ, the variation measure |Df| satisfies some invariance. We have

    Rng(σf)d|Df|=Sn1g(u)dS(f,u)=Sn1g(u)dSp(¯f,u), (4.12)

    for every g:RnR 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 KKno,

    V(K)(np)/pV(ΠpK)ωn/pn,

    with equality if and only if K is an origin-symmetric ellipsoid.

    Now, we prove Theorem 1.1.

    Proof. Let fBV(Rn), f0, p>1, pn and 1p+1q=1. By (2.3) and Remark 4.12, we calculate

    V(Πp¯f)=1nSn1hnΠp¯fdu=1nSn1(1cn2,pnωnSn1|uv|pdSp(¯f,v))npdu=1nSn1(1cn2,pnωnRn|uσf|pd|Df|)npdu=(cn2,pnωn)np1nSn1(Rn|uσf|pd|Df|)npdu. (5.1)

    Using Corollary 4.11, we have

    fnnn1(Dfn)nqV(¯f)npp, (5.2)

    where equality holds if and only if f=λ1B for some λRn and some Euclidean ball B. Thus, it follows from (5.2) and (5.1) that

    Sn1(Rn|uσf|pd|Df|)npduV(¯f)nppV(Πp¯f)fnnn1Dfnq(1n)n1(1cn2,pωn)np,

    i.e.,

    Sn1( Rn|uσf|pd|Df|)npduV(¯f)nppV(Πp¯f)ωn/pn(Dff1)n(1n)n1(1cn2,p)np, (5.3)

    where 1=n/(n1) and equality holds if and only if f=λ1B for some λRn and some Euclidean ball B. Applying Theorem 5.1, we see

    V(¯f)nppV(Πp¯f)ωn/pn1, (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

    Sn1( Rn|uσf|pd|Df|)npdu(1n)n1(1cn2,p)np(Dff1)n. (5.5)

    If f=λ1B for some λRn and some Euclidean ball B, then ¯f=λ1nprn1npB1(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 λRn and some Euclidean ball B.

    In summary, the equality holds in (5.5) if and only if f=λ1B for some λRn and some Euclidean ball B.

    Now, let pn in the inequality (5.5). It follows from the dominated convergence theorem that

    Sn1( Rn|uσf|nd|Df|)1du(1n)n1(1cn2,n)(Dff1)n. (5.6)

    Since

    Sn1( Rn|uσf|pd|Df|)npdu=(1n)n1(1cn2,p)np(Dff1)n

    with pn and f=λ1B for some λRn and some Euclidean ball B, we have

    Sn1( Rn|uσf|nd|Df|)1du=(1n)n1(1cn2,n)(Dff1)n

    by pn 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 fBV(Rn). Then

    nω1/nnf1Df, (5.7)

    where equality holds if and only if f=λ1B for some λRn and some Euclidean ball BRn.

    Firstly, we deduce the Lp Cauchy surface area formula, which is a direct consequence by the following lemma.

    Lemma 5.3. Let K,LKno and p1. Then

    Vp(L,ΠpK)=Vp(K,ΠpL). (5.8)

    Proof. Using (2.4) and Fubini's theorem, we directly calculate

    Vp(L,ΠpK)=1nSn1hpΠpK(u)dSp(L,u)=1nSn11cn2,pnωnSn1|uv|pdSp(K,v)dSp(L,u)=1nSn11cn2,pnωnSn1|uv|pdSp(L,u)dSp(K,v)=1nSn1hpΠpL(v)dSp(K,v)=Vp(K,ΠpL).

    The Lp Cauchy surface area formula is the following.

    Theorem 5.4. Let p1 and KKno. Then

    Sp(K)=1cn2,pnωnSn1(Sn1|uv|pdSp(K,v))du.

    Proof. Let p1 and KKno. 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)=Sn1hpΠpK(u)dSp(B1(0),u)=Sn1hpΠpK(u)du=1cn2,pnωnSn1(Sn1|uv|pdSp(K,v))du.

    Remark 5.5. Let p=1 in Theorem 5.4 and note that 2ω2ωn2=nωn. We get the classical Cauchy surface area formula, which reads as follows: If KKn, then

    S(K)=1ωn1Sn1Hn1(K|u)du.

    Now, we prove Corollary 5.2 by using Theorem 1.1 for fixed p>1 with pn.

    Proof. Let fBV(Rn). If f=0, then the inequality (1.1) holds trivially.

    Assume f0 and p>1. From Remark 4.12 and Theorem 5.4, we have

    cn2,pnωn=Sn1(1Sp(¯f)Sn1|uv|pdSp(¯f,v))du=Sn1(1DfRn|uσf|pd|Df|)du=Sn1( Rn|uσf|pd|Df|)du. (5.9)

    We set β=nn+p, that is, pn+1β=1. It follows from the H¨older inequality that

    Sn1( Rn|uσf|pd|Df|)du(Sn1( Rn|uσf|pd|Df|)npdu)pn(nωn)(n+p)/n,

    where equality holds if and only if

    Rn|uσf|pd|Df|=constant,

    that is, Πp¯f is a Euclidean ball with center at o. Thus, combining with (5.9), we see that

    Sn1( Rn|uσf|pd|Df|)npdu(1cn2,p)npnωn.

    Now, using Theorem 1.1, we calculate

    (1n)n1(1cn2,p)np(Dff1)nSn1( Rn|uσf|pd|Df|)npdu(1cn2,p)npnωn,

    that is,

    Dfnω1/nnf1.

    If f=λ1B for some λRn and some Euclidean ball BRn, then the equality holds in the inequality (1.7) and Πp¯f is a Euclidean ball with center at o. Thus,

    (1n)n1(1cn2,p)np(Dff1)n=Sn1( Rn|uσf|pd|Df|)npdu=(1cn2,p)npnωn,

    i.e.,

    Df=nω1/nnf1.

    If pn and fλ1B for all λRn and all Euclidean balls BRn, then

    (1n)n1(1cn2,p)np(Dff1)n>Sn1( Rn|uσf|pd|Df|)npdu.

    Thus,

    Df>nω1/nnf1.

    In summary, for pn we have that

    Df=nω1/nnf1

    if and only if f=λ1B for some λRn and some Euclidean ball BRn.

    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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