Loomis-Whitney inequalities in Heisenberg groups
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Loomis-Whitney inequalities in Heisenberg groups © The Author(s) 2022 Published version Fässler, Katrin; Pinamonti, Andrea Fässler, K., & Pinamonti, A. (2022). Loomis-Whitney inequalities in Heisenberg groups. Mathematische Zeitschrift, 301(2), 1983-2010. https://doi.org/10.1007/s00209-022-02968-y 2022
Mathematische Zeitschrift https://doi.org/10.1007/s00209-022-02968-y Mathematische Zeitschrift Loomis–Whitney inequalities in Heisenberg groups Katrin Fässler1,2 ·Andrea Pinamonti1,2 Received: 22 April 2021 / Accepted: 4 January 2022 © The Author(s) 2022 Abstract This note concerns Loomis–Whitney inequalities in Heisenberg groups Hn: |K| 2n j=1|πj(K)|n+1 n(2n+1),K⊂Hn. Here πj,j=1,...,2n,arethevertical Heisenberg projections to the hyperplanes {xj=0}, respectively, and |·|refers to a natural Haar measure on either Hn, or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group H1is a direct consequence of known Lpimproving properties of the standard Radon transform in R2. In this note, we show how the Loomis–Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in H1. The same approach, combined with multilinear interpolation, also yields the following strong type bound: Hn 2n j=1 fj(πj(p)) dp 2n j=1fjn(2n+1) n+1 for all nonnegative measurable functions f1,..., f2non R2n. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in Hn, we mention the following sharper version of the classical geometric Sobolev inequality in Hn: u2n+2 2n+1 2n j=1Xju1 2n,u∈BV(Hn), K. Fässler is supported by the Academy of Finland via the project Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups, grant Nos. 321696, 328846. A. Pinamonti is partially supported by supported by the University of Trento and GNAMPA of INDAM. BAndrea Pinamonti [email protected] Katrin Fässler [email protected] 1Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box. 35 (MaD), 40014 Jyväskylä, Finland 2Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo, Italy 123
K. Fässler, A. Pinamonti where Xj,j=1,...,2n, are the standard horizontal vector fields in Hn. Finally, we also establish an extension of the Loomis–Whitney inequality in Hn, where the Heisenberg vertical coordinate projections π1,...,π 2nare replaced by more general families of mappings that allow us to apply the same inductive approach based on the L3/2-L3boundedness of an operator in the plane. Keywords Radon transform ·Loomis–Whitney inequality ·Heisenberg group ·Sobolev inequality ·Isoperimetric inequality Mathematics Subject Classification 28A75 primary; 52C99 ·46E35 ·35R03 secondary 1 Introduction The Loomis–Whitney inequality in Rdbounds the volume of a set K⊂Rdby the areas of its coordinate projections: |K|≤ d j=1|˜πj(K)|1 d−1,(1.1) where ˜πj(x1,...,xd)=(x1,...,xj−1,xj+1,...,xd).Here|A|refers to k-dimensional Lebesgue outer measure in Rkwhenever A⊂Rk. The inequality (1.1) is due to Loomis and Whitney [37] from 1949. It is trivial for d=2 and follows by induction, using Hölder’s inequalities, for d>2. The Loomis–Whitney inequality is one of the fundamental inequalities in geometry and has been studied intensively; we refer to [6,8,12,25,33] and references therein for a historical account and some applications of the Loomis–Whitney inequality. The present note discusses analogues of (1.1) in Heisenberg groups Hn.Itaroseasa complement to manuscript [23] with Tuomas Orponen, in which we reduced the proof of the Loomis–Whitney inequality for H1to an incidence geometric problem in the plane that we resolved using the method of polynomial partitioning. Later we learned that the Loomis– Whitney inequality in the first Heisenberg group—and inequalities of similar type—had already been obtained earlier [18,19,32,38] by a Fourier-analytic approach or the so-called method of refinements, albeit not phrased in terms of Heisenberg projections. In addition to acknowledging previous work, the aim of the present note is to show how the Loomis– Whitney inequality in Hnfor n>1 can be proven by induction, similarly as the original inequality [37], but now using the version in H1as a base case. Alternatively, one could apply the method of refinements also for n>1, see the related comment in [42,Sect.4].The inductive approach in the present note has the advantage of easily yielding certain strong-type endpoint inequalities, see Theorem 1.8, which are not covered by [42] or other literature we are aware of. For applications to geometric Sobolev and isoperimetric inequalities in Hn,the weak-type inequalities would however be sufficient. 1.1 Heisenberg groups The nth Heisenberg group Hnis the group (R2n+1,·)with (x,t)·(x,t):= ⎛ ⎝x+x,t+t+1 2 n j=1 xjx n+j−xn+jxj⎞ ⎠,(1.2) 123
Loomis–Whitney inequalities in Heisenberg... which makes it a nilpotent Lie group of step 2. Here, (x,t)denotes a point in R2n+1with x=(x1,...,x2n)∈R2nand t∈R.Forx∈R2nand k∈{1,...,2n}, we will use the symbol ˆxkto denote either the point in R2nthat is obtained by replacing the k-th coordinate of xwith 0, or the point in R2n−1that is obtained by simply deleting the k-th coordinate of x. The meaning should always be clear from the context. In geometric measure theory of the sub-Riemannian Heisenberg group [41], an important role is played by Heisenberg projections that are adapted to the group and dilation structure of Hnand that map onto homogeneous subgroups of Hn. We only consider projections associated to the "coordinate" hyperplanes containing the t-axis, so we limit our discussion to those. Let Wj⊂Hn,j=1,...,2n,bethe(1-codimensional)vertical subgroups of Hn given by the hyperplanes {(x,t)∈R2n+1:xj=0}, respectively. Write Lj:= {(0,...,0,xj,0,...,0):xj∈R} for the span of the j-th standard basis vector. So Ljis a complementary (1-dimensional) horizontal subgroup of Wj. This means, for example, that every point p∈Hnhas a unique decomposition p=wj·lj,wherewj∈Wjand lj∈Lj. These decompositions give rise to the vertical coordinate projections p→ wj=: πj(p)∈Wj,j=1,...,2n. Using the group product in (1.2), it is easy to write down explicit expressions for πj: πj(x,t)=(ˆxj,t+xjxn+j 2)and πn+j(x,t)=(ˆxn+j,t−xjxn+j 2), j=1,...,n. (1.3) Readers who are not comfortable with the Heisenberg group can simply identify Wjwith R2n, and consider the maps (x,t)→ (x1,...,xj−1,xj+1,...,x2n,t+xjxn+j 2), for j=1,...,n, and their analogs for j=n+1,...,2n, without paying attention to their origin. It is clear that the projections π1,...,π 2nare smooth, and hence locally Lipschitz with respect to the Euclidean metric in R2n+1, and they satisfy det Dπj(p)Dπj(p)t≥1,j=1,...,2n,p∈R2n+1.(1.4) Vertical projections are, in fact, not Lipschitz with respect to the Korányi distance d(p,q)= q−1·pon Hn. Nonetheless they play a significant role in the geometric measure theory of Heisenberg groups—as do orthogonal projections in Rd—so they have been actively investigated in recent years, see [2,3,15,22,34,35]. The vertical projections are non-linear maps, but their fibres π−1 j{w}are nevertheless lines. In fact, the fibres of πjare precisely the left translates of the line Lj,thatis,π−1 j{w}=w·Ljfor w∈Wj. For subsets of Hn∼ =R2n+1, the notation |·|will refer to Lebesgue (outer) measure on R2n+1, and for subsets of a vertical plane R2n∼ =Wj⊂Hn, the notation |·|will refer to Lebesgue (outer) measure in R2n. Up to multiplicative constants, they could also be defined as the (2n+2)-and(2n+1)-dimensional Hausdorff measures, respectively, relative to the Korányi metric on Hn. So, our measures coincide with canonical "intrinsic" objects in Hn. All integrations on Hnor Wjwill be performed with respect to Lebesgue measures. 123
K. Fässler, A. Pinamonti 1.2 Loomis–Whitney inequalities in Hnand their generalizations We can now state a variant of the Loomis–Whitney inequality (1.1) for subsets of Hnin terms of the vertical coordinate projections πj.InRd, the inequality makes a reference to the d orthogonal coordinate projections π1,...,πd. These are, now, best viewed as the projections whose fibres are translates of lines parallel to the coordinate axes. In Hn, we consider instead the vertical projections πjwhose fibres are left translates of Lj,j=1,...,2n; the precise formulae were stated in (1.3). With this notation, the following variant of the Loomis–Whitney inequality holds: Theorem 1.5 (Loomis–Whitney inequality in Hn)Fix n ∈N.LetK ⊂R2n+1(or K ⊂Hn) be an arbitrary set. Then |K| 2n j=1|πj(K)|n+1 n(2n+1).(1.6) Here and in the following, the symbol indicates that the inequality holds up to a positive and finite multiplicative constant on the right-hand side. We only have to prove the inequality for Lebesgue measurable sets K⊂R2n+1. In the general case, we simply pick Gδ-sets Kj⊂R2nwith Kj⊇πj(K)and |Kj|=|πj(K)|for j=1,...,2n, assuming that the right-hand side of (1.6) is finite. Then K:= 2n j=1π−1 j(Kj)is a Lebesgue measurable subset of R2n+1that contains Kand it suffices to apply the Loomis–Whitney inequality to K. So we consider only Lebesgue measurable sets Kin the following. By the inner regularity of the Lebesgue measure, Theorem 1.5 is then equivalent to the validity of (1.6)forall compact sets K⊂R2n+1. Since every such set satisfies χK(p)≤2n j=1χπj(K)(πj(p)), for all p∈R2n+1, and on the other hand, 2n j=1π−1 j(Kj)is compact in R2n+1whenever K1,...,K2nare compact subsets of R2n, Theorem 1.5 is equivalent to the statement that R2n+1 2n j=1 χKj(πj(p)) dp 2n j=1|Kj|n+1 n(2n+1)(1.7) holds for all compact sets K1,...,K2n⊂R2n. Here we have identified, for j=1,...,2n, the {xj=0}-plane in R2n+1with R2n,sothatπ1,...,π 2nare now mappings from R2n+1 to R2n. Using this expression, it is evident that Theorem 1.5 follows from the next result: Theorem 1.8 Fix n ∈N.Then R2n+1 2n j=1 fj(πj(p)) dp 2n j=1fjn(2n+1) n+1 ,(1.9) for all nonnegative Lebesgue measurable functions f1,..., f2non R2n. The coarea formula coupled with (1.4) shows that the preimages of Lebesgue null sets in R2n under πjare Lebesgue null sets in R2n+1,andso fj◦πj:R2n+1→[0,+∞] is Lebesgue measurable under the assumptions of the theorem, and the integral on the left-hand side of (1.9) makes sense. The bilinear case (n=1) of Theorem 1.8 follows directly from the L3/2−L3boundedness of the standard Radon transform in R2, and as such was known—by a Fourier-analytic proof— at least since the work of Oberlin and Stein [38]; see Sect. 2. Theorem 1.8 for n=1isalso 123
Loomis–Whitney inequalities in Heisenberg... an instance of [18, Theorem 1.1] (with b=(2,2)in [18, (1.6)] and (p1,p2)=(3/2,3/2)in [18, (1.8)]). The corresponding weak-type bound (Theorem 1.5 for n=1) was also obtained by Gressman as a special case of the endpoint restricted weak-type estimates in [32, Theorem 2]. Due to the nilpotent group structure of the Heisenberg group and the invariance of the problem under Heisenberg dilations, it is a particularly simple instance of Gressman’s more general theorem. The proofs in [18,32]usedanadaptationofthemethod of refinements,which was initiated by Christ [16] in order to prove Lp−Lqbounds for certain convolution-type operators. To the best of our knowledge, Theorem 1.8 for n>1 has not appeared in the literature before. Stovall proved in [42] similar inequalities for multilinear Radon-like transforms, but (1.9)forn>1 constitutes a strong-type endpoint case that is not covered by her work. In her notation, our setting corresponds to b(p)=((n+1)/n,...,(n+1)/n), which is a point on the boundary of the polytope Pmentioned in [42, Theorem 3]. Our approach to Theorem 1.8 can be applied to prove something a bit more general, see Theorem 5.16 for the precise statement. The idea is to apply the same inductive procedure and reduce the claim to an L3/2-L3boundedness statement for a certain operator in the plane. In the case of Theorem 1.8, this operator happens to be the standard Radon transform, but other choices are possible as well, for instance convolution by a fixed parabola in R2,cf.the use of (5.7) in connection with Example 5.4. It is easy to see that the exponents in the Heisenberg Loomis–Whitney inequality (1.6) are sharp by considering boxes of the form [−r,r]2n×[−r2,r2]. Besides the difference in the definition of the projections ˜πjand πj, there is another obvious difference between (the case d=2n+1 of) the standard Loomis–Whitney inequality (1.1), and (1.6): the former bounds the volume of Kin terms of 2n+1 projections, and the latter in terms of only 2n projections. One might therefore ask: is there a version of (1.1)for2northogonal projections R2n+1→R2n—and does it look like (1.6)? The answer is negative. This is a very special case of [5, Theorem 1.13] (cf. also [20,42,43]), but perhaps it is illustrative to see an explicit computation for n=1: Example 1.10 Consider the two standard orthogonal coordinate projections ˜π1,˜π2in R3to the x2t-andx1t-planes. If K=[0,1]2×[0,δ],then|K|=δ,andalso|˜π1(K)|=δ=|˜π2(K)|. So, for δ>0 small, an inequality of the form |K||˜π1(K)|λ·|˜π2(K)|λ(1.11) can only hold for λ≤1 2. On the other hand, if KR=[0,R]3, with R1, then |KR|=R3 and |˜π1(KR)|=R2=|˜π2(KR)|,so(1.11) can only hold for λ≥3 4. The latter example naturally does not contradict (1.6): note that |πj(KR)|∼R3for R1. 1.3 Gagliardo–Nirenberg–Sobolev inequalities in Hn In Rd, it is well-known that the Loomis–Whitney inequality implies the Gagliardo– Nirenberg–Sobolev inequality fd/(d−1)≤ d j=1∂jf1/d 1,f∈C1 c(Rd). (1.12) Similarly, an Hn-analogue of (1.12) can be obtained as a corollary of Theorem 1.5: 123
K. Fässler, A. Pinamonti Theorem 1.13 Let f ∈BV(H). Then, f2n+2 2n+1 2n j=1Xjf1 2n.(1.14) Here Xj=∂xj−xn+j 2∂tand Xn+j=∂xn+j+xj 2∂t,(j=1,...,n), (1.15) are the standard left-invariant "horizontal" vector fields in Hn,andBV(Hn)refers to functions f∈L1(Hn)whose distributional Xjderivatives are signed Radon measures with finite total variation, denoted ·. Theorem 1.13 presents a sharper version of the well-known "geometric" Sobolev inequality f2n+2 2n+1∇Hf,f∈BV(Hn), (1.16) proven by Pansu [40]forn=1 as a corollary of the isoperimetric inequality in H1.Here ∇Hf=(X1f,...,X2nf). Versions of geometric Sobolev inequalities and isoperimetric inequalities were obtained in Hnand even more general frameworks by several authors, for instance in [14,30]. A proof of (1.16)forn=1, using the fundamental solution of the sub-Laplace operator H, is discussed in [13, Sect. 5.3], following the approach of [14]. On the other hand, Theorem 1.13 can be derived from Theorem 1.5. This deduction follows a standard argument, but we present it here to highlight the fact that the geometric Sobolev and isoperimetric inequalities in all Heisenberg groups are ultimately based on planar geometry and they can be deduced from boundedness properties of the Radon transform in R2. Theorem 1.8 is related to Brascamp-Lieb inequalities. We direct the reader to e.g. [4,5,10] and the references therein. Euclidean Loomis–Whitney and Brascamp-Lieb inequalities can be proven by the technique of heat flow monotonicity,see[5]. The same approach has been attempted in Carnot groups by Bramati [9], but there seems to be a gap in the argument, which has been confirmed with the author. More precisely, the exponents appearing in the proof of [9, Theorem 3.2.3] have not been chosen consistently. It remains an open problem to see whether the Loomis–Whitney inequalities in Carnot groups can be obtained by the heat flow approach. Structure of the paper. In Sect. 2, we explain how Theorems 1.5 and 1.8 for n=1 follow from known Lpimproving properties of the Radon transform in R2.InSect.3, we deduce Theorems 1.5 and 1.8 for arbitrary n>1 by induction from the corresponding inequalities in H1. In Sect. 4, we show how to derive the Gagliardo–Nirenberg–Sobolev inequality, Theorem 1.13, as an application of the Loomis–Whitney inequality in Hn. Finally, in Sect. 5we explain how to adapt the approach from Sect. 3to prove the generalized Loomis–Whitney-type inequality stated in Theorem 5.16. 2 Inequalities in the first Heisenberg group In this section, we review the proof for the Loomis–Whitney inequality in the first Heisenberg group. For this purpose it is more convenient to use slightly different notation. In particular, points in R3will be denoted by (x,y,t)(instead of (x,t)=(x1,x2,t)). The group product of H1then reads in coordinates as follows: (x,y,t)·(x,y,t):= (x+x,y+y,t+t+1 2(xy−yx)). (2.1) 123
Loomis–Whitney inequalities in Heisenberg... The vertical Heisenberg projections to the yt-andthext-plane, respectively, are explicitly given by π1(x,y,t)=(0,y,t+xy 2)and π2(x,y,t)=(x,0,t−xy 2). We recall the statement of Theorems 1.5 and 1.8 for n=1: Theorem 2.2 (Loomis–Whitney inequality in H1)Let K ⊂H1be arbitrary. Then, |K||π1(K)|2/3·|π2(K)|2/3.(2.3) Theorem 2.4 For all nonnegative Lebesgue measurable functions f1and f2on R2it holds that R3 f1(π1(p)) f2(π2(p)) dp f13 2f23 2.(2.5) On the left-hand side of (2.3), the notation "|·|" refers to Lebesgue outer measure on R3. Similarly, on the right-hand side of (2.3), the notation "|·|" refers to Lebesgue outer measure on R2. Clearly, Theorem 2.4 implies Theorem 2.2. We now explain how Theorem 2.4 itself follows directly from known Lp-improving properties of the standard Radon transform in the plane R2. Let S1be the unit sphere in R2. For a smooth, compactly supported function fon R2,the Radon transform (or X-ray transform) Rf is defined by Rf(σ, s):= z,σ =s f(z)dz,(σ,s)∈S1×R.(2.6) Here dz is the 1-dimensional Lebesgue measure on the line {z∈R2:z,σ=s}.Using Fourier analysis (notably Plancherel’s theorem) and complex interpolation, Oberlin and Stein [38] proved that Rextends to a bounded operator from L3/2(R2)to L3(S1×R). Their result is more general, but this is the only information one needs to deduce Theorem 2.4. The connection between inequality (2.5) and the Radon transform is illustrated by the formula R3 f1(π1(p)) f2(π2(p)) dp =R2 R(f1)(σ(x), sx,t)f2(x,t)d(x,t) √1+x2(2.7) with sx,t=t/√1+x2and σ(x):= 1 √1+x2(−x,1)for smooth compactly supported functions f1and f2on R2. The proof of inequality (2.5) using the result in [38] is an instance of a more general phenomenon that relates Lp-improving properties of averaging operators along curves to inequalities of the form (2.5) with two factors in the integral. The general framework is explained in detail in [20, 9.5. Double fibration formulation] and [43, Sect. 1]. For our purpose it is convenient to work with a linear operator Tthat yields functions on R2, rather than S1×Ras in the case of the Radon transform, so instead of applying directly (2.7), we will pass via an identity of the form R3 f1(π1(p)) f2(π2(p)) dp =R2 Tf 1(x,t)f2(x,t)d(x,t); see the proof of Theorem 2.4. For smooth, compactly supported functions fon R2,wedefine Tf(x,t):= R f(y,t+xy)dy,(x,t)∈R2.(2.8) The next statement follows immediately from [38] by relating the operator Tto the Radon transform R, and we do not claim any novelty for it, see also [17,Sect.2]. 123
K. Fässler, A. Pinamonti Theorem 2.9 There exists a constant C such that the operator T defined in (2.8) satisfies Tf3≤Cf3 2 for all smooth, compactly supported functions f . Proof We reduce Theorem 2.9 to a statement about the Radon transform that was proven in [38]. We fix a smooth compactly supported function fand start by writing Tf3=R2R f(y,t+xy)dy 3 d(x,t)1 3 (2.10) =R2R f(y,t+xy)1+x2dy 3d(x,t) (1+x2)3/21 3 =⎡ ⎣R2x,t fdλx,t 3d(x,t) (1+x2)3/2⎤ ⎦ 1 3 .(2.11) Here dλx,tdenotes the 1-dimensional Lebesgue measure on the line x,t:= z∈R2:z,σ(x)= t √1+x2 ={(y,t+xy):y∈R}with σ(x):= 1 √1+x2−x 1. Thus, recalling the definition of the Radon transform in (2.6), we obtain from (2.11)that Tf3=R2|Rf(σ(x), sx,t)|3d(x,t) (1+x2)3/21 3 =RR|Rf(σ (x), sx,t)|3dt √1+x2dx 1+x21 3 with sx,t=t/√1+x2. Changing variables in the inner integral, and observing that x→ σ(x)parameterizes an arc in S1, we then deduce that Tf3=RR|Rf(σ(x), s)|3ds|σ(x)|dx1 3 ≤S1R|Rf(σ, s)|3dsdσ1 3 =Rf3, where σdenotes the usual Lebesgue (arc-length) measure on S1. Now the theorem follows from the inequality Rf3≤Cf3 2for the Radon transform, which was established as a special case of [38, Theorem 1]. Theorem 2.4 is an immediate corollary of Theorem 2.9. Proof of Theorem 2.4 It suffices to prove the theorem for nonnegative smooth, compactly supported functions on R2. Indeed, if f1is an arbitrary nonnegative Lebesgue measurable function on R2,wetakeasequence(f1,k)k∈Nof nonnegative C∞ cfunctions which converges to f1with respect to ·3/2and pointwise almost everywhere. In the same way, we approximate a 123
Loomis–Whitney inequalities in Heisenberg... After this transformation, the fn-term is independent of the n-th coordinate of x. We can separate it from the other factors by applying Hölder’s inequality with exponents p=2n and p=2n/(2n−1)to the expression inside the square brackets. This yields J≤R FnFdx2n(3.15) where Fn:= R2n−1 fn(ˆxn,t)2n+1d(ˆxn,2n,t)1 2n+1 and F:= ⎡ ⎢ ⎢ ⎢ ⎣R2n−1⎛ ⎜ ⎜ ⎝R 2n j=1 j=n,2n fj(πj(x,t−1 2xnx2n))2n+1 2ndxn⎞ ⎟ ⎟ ⎠ 2n 2n−1 d(ˆxn,2n,t)⎤ ⎥ ⎥ ⎥ ⎦ 2n−1 2n+1 . Applying once more Hölder’s inequality, but now to the x2n-integral in (3.15), and with exponents p=2n+1andp=(2n+1)/2n, yields J≤R F2n+1 ndx2n1 2n+1R F 2n+1 2n dx2n2n 2n+1 =Jn·J. Here Jn:= R F2n+1 ndx2n1 2n+1 =R2n fn(ˆxn,t)2n+1d(ˆxn,t)1 2n+1 =fn2n+1 is one of the factors in the desired upper bound for J, recall (3.14). Hence, in order to prove (3.14), it suffices to show that J:= R F 2n+1 2n dx2n2n 2n+1 f12n+1 2fn+12n+1 2 n−1 j=2fj2n+1fn+j2n+1.(3.16) To do so, we will finally use our induction hypothesis. We start by expanding J= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝R ⎡ ⎢ ⎢ ⎢ ⎢ ⎣R2n−1 ⎛ ⎜ ⎜ ⎜ ⎝R 2n j=1 j=n,2n fj(π j(x,t−1 2xnx2n)) 2n+1 2ndxn⎞ ⎟ ⎟ ⎟ ⎠ 2n 2n−1 d(ˆxn,2n,t)⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 2n−1 2n dx2n ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 2n 2n+1 . Applying Minkowski’s integral inequality inside the square brackets, then Fubini’s theorem and the transformation t→ τ=t−1 2xnx2nyields J≤⎛ ⎜ ⎜ ⎜ ⎝R2⎡ ⎢ ⎢ ⎣R2n−1 2n j=1 j=n,2n fj(πj(x,τ))2n+1 2n−1d(ˆxn,2n,τ) ⎤ ⎥ ⎥ ⎦ 2n−1 2n d(xn,x2n)⎞ ⎟ ⎟ ⎟ ⎠ 2n 2n+1 .(3.17) We recall that 123
K. Fässler, A. Pinamonti fj(πj(x,τ))=fj(ˆxj,τ +1 2xjxn+j), if j=1,...,n−1, fj(ˆxj,τ −1 2xj−nxj), if j=n+1,...,2n−1.(3.18) We will continue the upper bound for Jby applying the induction hypothesis to the expression inside the square brackets. To do so, we temporarily denote points in Hn−1in coordinates by (u,t)=(u1,...,u2n−2,τ). Here, uis a point in R2n−2, and similarly as before, ˆukdenotes the point in R2n−3that is obtained from uby deleting the k-th coordinate. To write the inner integral on the right-hand side of (3.17) in a form where the induction hypothesis is applicable, we fix xn,x2n∈Rand define the functions gxn,x2n,j, j∈{1,...,2n−2}on R2n−2: gxn,x2n,j(ˆuj,t) := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ f1(u2,...,un−1,xn,un,...,u2n−2,x2n,t)2n+1 2n−1,j=1, fj(u1,...,uj−1,uj+1,...,un−1,xn,un,...,u2n−2,x2n,t)2n+1 2n−1, 2≤j≤n−2 fn−1(u1,...,un−2,xn,un,...,u2n−2,x2n,t)2n+1 2n−1,j=n−1, (3.19) and gxn,x2n,j(ˆuj,t) := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ fn+1(u1,...,un−1,xn,un+1,...,u2n−2,x2n,t)2n+1 2n−1,j=n, fj+1(u1,...,un−1,xn,un,...,uj−1,uj+1,...u2n−2,x2n,t)2n+1 2n−1, n+1≤j≤2n−3, f2n−1(u1,...,un−1,xn,un,...,u2n−3,x2n,t)2n+1 2n−1,j=2n−2. (3.20) With this notation in place, and recalling (3.18), we can restate (3.17) equivalently as follows J≤⎛ ⎜ ⎝R2⎡ ⎣R2n−1 2n−2 j=1 gxn,x2n,j(πj(u,t)) d(u,t)⎤ ⎦ 2n−1 2n d(xn,x2n)⎞ ⎟ ⎠ 2n 2n+1 , where πjnow denotes the Heisenberg projection from Hn−1to the vertical plane {uj=0} (identified with R2n−2). The induction hypothesis applied to the inner integral yields J⎛ ⎜ ⎜ ⎜ ⎝R2⎡ ⎢ ⎢ ⎣gxn,x2n,12n−1 2gxn,x2n,n2n−1 2 2n−2 j=1 j/∈1,n gxn,x2n,j2n−1⎤ ⎥ ⎥ ⎦ 2n−1 2n d(xn,x2n)⎞ ⎟ ⎟ ⎟ ⎠ 2n 2n+1 . (3.21) Next we apply the multilinear Hölder inequality with exponents p1=pn=nand p2=...=pn−1=pn+1=...=p2n−2=2n. Note that 2n−2 j=1 1 pj=2 n+2n−4 2n=1, 123
Loomis–Whitney inequalities in Heisenberg... as desired. Hence we deduce from (3.21)that JR2gxn,x2n,1 2n−1 2 2n−1 2 d(xn,x2n)2 2n+1R2gxn,x2n,n 2n−1 2 2n−1 2 d(xn,x2n)2 2n+1 · 2n−2 j=1 j/∈1,nR2gxn,x2n,j2n−1 2n−1d(xn,x2n)1 2n+1 . Recalling the definition of gxn,x2n,jfor j=1,...,2n−2 as stated in (3.19)and(3.20), we obtain immediately Jf12n+1 2fn+12n+1 2 n−1 j=2fj2n+1fn+j2n+1. as desired; recall (3.16). This proves (3.14) and thus establishes the statement about k=1 in the induction claim (3.2)forn. The other values of kare treated analogously, and hence we have established (3.2). 4 Applications of the Loomis–Whitney inequalities in Heisenberg groups In this section, we derive the Gagliardo–Nirenberg–Sobolev inequality in Hn, and its variant Theorem 1.13, from the Loomis–Whitney inequality, Theorem 1.5. As a corollary of Theorem 1.13, we obtain the isoperimetric inequality in Hn(with a non-optimal constant). At the end of the section, we also show how the Loomis–Whitney inequality can be used, directly, to infer a variant of the isoperimetric inequality, without passing through the Sobolev inequality. The arguments presented here are very standard ( [1,29,37]), and we claim no originality. A version of this section, in the context of the first Heisenberg group, was already contained in our joint work [23] with Tuomas Orponen. In his thesis [9], Bramati also gave an argument to deduce the Gagliardo–Nirenberg–Sobolev and isoperimetric inequalities in H1from the strong version of the Loomis–Whitney inequality stated in Theorem 2.4. We start by recalling the statement of Theorem 1.13: Theorem 4.1 Let f ∈BV(Hn). Then, f2n+2 2n+1 2n j=1Xjf1 2n.(4.2) Recall that f∈BV(Hn)if f∈L1(Hn), and the distributional derivatives Xjf, j=1,...,2n, are finite signed Radon measures. Smooth compactly supported functions are dense in BV(Hn)in the sense that if f∈BV(Hn), then there exists a sequence {ϕk}k∈N⊂C∞ c(R2n+1)such that ϕk→falmost everywhere (and in L1(Hn)if desired), and Zϕk→Zffor Z∈{X1,...,X2n}. For a reference, see [27, Theorem 2.2.2]. With this approximation in hand, it suffices to prove Theorem 4.1 for, say, f∈C1 c(R2n+1).The following lemma contains most of the proof: Lemma 4.3 Let f ∈C1 c(R2n+1), and write Fk:= {p∈R2n+1:2k−1≤|f(p)|≤2k},k∈Z.(4.4) 123
K. Fässler, A. Pinamonti Then, |πj(Fk)|≤2−k+2Fk−1|Xjf|,j=1,...2n.(4.5) Proof By symmetry, it suffices to prove the inequality in (4.5)for j=1,...,n.Letw= (ˆxj,t)∈πj(Fk), denote by ejthe j-th unit vector, and fix p=w·xjej∈Fksuch that πj(p)=w.Inparticular,|f(p)|≥2k−1. Recall the notation Lj=span(ej)={xjej: xj∈R}and the definition of ˆxjgiven below (1.2). Since fis compactly supported, we may pick another point p∈w·Ljsuch that f(p)=0. Since |f|is continuous, we infer that there is a non-degenerate line segment Ion the line w·Ljsuch that 2k−2≤|f(q)|≤2k−1 for all q∈I(hence I⊂Fk−1), and |f|takes the values 2k−2and 2k−1, respectively, at the endpoints qi=w·xj,iejof I,i∈{1,2}.Defineγ(xj):= w·xjej=(x,t−1 2xjxn+j). With this notation, 2k−2≤|f(q1)−f(q2)|≤xj,2 xj,1|(f◦γ) (xj)|dxj ≤{xj:(x,t−1 2xjxn+j)∈Fk−1}|Xjf(x,t−1 2xjxn+j)|dy. Writing (x,t):= (ˆxj,t)·xjej=(x,t−1 2xjxn+j), and integrating over (x1,...,xj−1,xj+1,...,x2n,t)=(ˆxj,t)∈πj(Fk)⊂Wj, it follows that 2k−2|πj(Fk)|≤πj(Fk){xj:(x,t)∈Fk−1}|Xjf((x,t))|dxjdˆxjdt.(4.6) Finally, we note that J=det D≡1. Therefore, using Fubini’s theorem, and performing a change of variables to the right-hand side of (4.6), we see that 2k−2|πj(Fk)|≤{(x,t)∈R2n+1:(x,t)∈Fk−1}|Xjf((x,t))|d(x,t) =Fk−1|Xjf(x,t)|d(x,t). This completes the proof. We are then prepared to prove Theorem 4.1: Proof of Theorem 4.1 Fix f∈C1 c(R2n+1), and define the sets Fk,k∈Z,asin(4.4). Using first Theorem 1.5, then Lemma 4.3, then the generalized Hölder’s inequality with p1=...= p2n=2n, and finally the embedding 1→(2n+2)/(2n+1), we estimate as follows: |f|2n+2 2n+1∼ k∈Z 2(2n+2)k 2n+1|Fk| k∈Z 2(2n+2)k 2n+1 2n j=1|πj(Fk)|n+1 n(2n+1) k∈Z 2n j=1Fk−1|Xjf|n+1 n(2n+1) 123
Loomis–Whitney inequalities in Heisenberg... 2n j=1' k∈ZFk−1|Xjf|2n+2 2n+1(1 2n 2n j=1' k∈ZFk−1|Xjf|(2n+2 2n(2n+1)∼ 2n j=1Xjf 2n+2 2n(2n+1) 1. Raising both sides to the power (2n+1)/(2n+2)completes the proof. We conclude the section by discussing isoperimetric inequalities. A measurable set E⊂ Hnhas finite horizontal perimeter if χE∈BV(Hn).HereχEis the characteristic function of E. Note that our definition of BV(Hn)implies, in particular, that |E|<∞.Wefollow common practice, and write PH(E):= ∇HχE. For more information on sets of finite horizontal perimeter, see [26]. Now, applying Theorem 4.1 to f=χE, we recover the following isoperimetric inequality (with a non-optimal constant): Theorem 4.7 There exists a constant C >0such that |E|2n+1 2n+2≤CP H(E)(4.8) for any measurable set E ⊂Hnof finite horizontal perimeter. For n=1, this is Pansu’s isoperimetric inequality [40], which has later been generalized to Hnand beyond [14,30]. We remark that the aprioriassumption |E|<∞is critical here; for example the theorem evidently fails for E=Hn, for which |E|=∞but ∇HχE=0. We conclude the paper by deducing a weaker version of (4.8)(even)moredirectlyfromthe Loomis–Whitney inequality. Namely, we claim that |E|2n+1 2n+2≤CH2n+1 d(∂ E)(4.9) for any bounded measurable set E⊂Hn,whereH2n+1 ddenotes the 2n+1-dimensional Hausdorff measure on Hnwith respect to the Korányi distance (or the standard left-invariant sub-Riemannian metric). This inequality is, in general, weaker than (4.8): at least for open sets E⊂Hn, the property H2n+1 d(∂ E)<∞implies that PH(E)<∞,andthenPH(E) H2n+1 d(∂ E),see[28, Theorem 4.18]. However, if Eis a bounded open set with C1boundary, then H2n+1 d(∂ E)∼PH(E),see[26, Corollary 7.7]. To prove (4.9), we need the following auxiliary result, see [15, Lemma 3.4] and [24, Remark 4.7]: Lemma 4.10 Let n ∈N. There exists a constant Cn>0such that the following holds. Let W⊂Hnbe a vertical subgroup of codimension 1. Then, |πW(A)|≤CH2n+1 d(A), A⊂Hn.(4.11) Proof of (4.9) Let E⊂Hbe bounded and measurable. We first claim that πj(E)⊆πj(∂ E), j=1,...,2n.(4.12) Let w∈πj(E)and consider π−1 j{w}=w·Ljwhere Ly=span(ej). By definition there exists xj,1∈Rsuch that w·xj,1ej∈Eand since Eis bounded there also exists xj,2∈R such that w·xj,2∈Hn\E.Sincew·Ljis connected, there finally exists xj,3∈Rsuch 123
K. Fässler, A. Pinamonti that w·xj,3ej∈∂Ewhich immediately implies (4.12). Using Theorem 1.5 and (4.12), we get |E| 2n j=1|πj(∂ E)|n+1 n(2n+1). Now the isoperimetric inequality (4.9) follows using Lemma 4.11. 5 Generalized Loomis–Whitney inequalities by induction The approach described in Sect. 3can be used to prove something a bit more general, namely we can replace the vertical Heisenberg projections π1,...,π 2nby projection-type mappings of the form ρj:R2n+1→R2n,ρ j(x,t)=(ˆxj,t+hj(x)), j=1,...,2n,(5.1) for suitable C1maps hj:R2n→R. The precise condition is stated in Definition 5.2 and it is tailored so that a Loomis–Whitney-type inequality for ρ1,...,ρ 2ncan be established basedontheL3/2-L3boundedness of a linear operator in the plane, analogously as we did for π1,...,π 2nand the Radon transform in Sects. 2-3. By a simple change-of-variables, one can generalize the setting even slightly further, see Remark 5.18. For arbitrary C1functions hj, the mappings ρjdefined in (5.1) satisfy a condition analogous to (1.4)forπj, which ensures by the coarea formula that the preimage of a Lebesgue null set in R2nunder ρjis a Lebesgue null set in R2n+1.Moreprecisely,wehave det(DρjDρt j)=det ⎛ ⎜ ⎜ ⎜ ⎝ 1 ...∇ˆxjh 1 ∇ˆxjh(1+|∇h|2) ⎞ ⎟ ⎟ ⎟ ⎠=1+(∂xjh)2. By the reasoning below Theorem 1.8 it follows that f◦ρjis Lebesgue measurable on R2n+1 if fis Lebesgue measurable on R2n. Definition 5.2 (L3/2-L3property) We say that a family {h1,...,h2n}of C1functions hjon R2nhas the L3/2-L3property if there exists a constant C<∞such that the following holds for all k=1,...,n: •If n>1, then for every ˆxk,n+k∈R2n−2, the operator Tk,ˆxk,n+k,definedby Tk,ˆxk,n+kf(xk,t):= R f(xn+k,t+hk(x)−hn+k(x)) dxn+k,f∈C∞ c(R2) satisfies Tk,ˆxk,n+kf3≤Cf3 2,f∈C∞ c(R2). Here, the coordinates of ˆxk,n+k∈R2n−2are xi,i∈{1,...,2n}\{k,n+k},and x=(x1,...,xk,...,xn+k,...,x2n). •If n=1, then the operator T1=T,definedby Tf(x1,t):= R f(x2,t+h1(x1,x2)−h2(x1,x2))dx2,f∈C∞ c(R2) 123
Loomis–Whitney inequalities in Heisenberg... satisfies Tf3≤Cf3 2,f∈C∞ c(R2). We next give examples of functions {h1,...,h2n}with the properties stated in Definition 5.2. Essentially, for k=1,...,n,wetakehkand hn+kto be polynomials of second degree as functions of xkand xn+kso that Theorem 5.5 is applicable. This class of examples includes the functions hj(x)=)1 2xjxn+j,j=1,...,n, −1 2xj−nxj,j=n+1,...,2n. (5.3) associated to the standard Heisenberg vertical coordinate projections ρj=πj,j= 1,...,2n. Example 5.4 Fix n>1, bj∈Rand cj,a∈C1(R2n−2)for j=1,...,2nand multi-indices a∈A:= {(0,0), (1,0), (0,1), (2,0), (0,2)}.Fork=1,...,n,wedefine hk(x):= bkxkxn+k+ a=(a1,a2)∈A ck,a(ˆxk,n+k)xa1 kxa2 n+k and hn+k(x):= bn+kxkxn+k+ a=(a1,a2)∈A cn+k,a(ˆxk,n+k)xa1 kxa2 n+k. Then the operators appearing in Definition 5.2 are given by Tk,ˆxk,n+kf(xk,t):= R fxn+k,t+Hk,n+k(x)dxn+k,f∈C∞ c(R2), where Hk,n+k(x):= (bk−bn+k)xkxn+k+ a=(a1,a2)∈A*ck,a(ˆxk,n+k)−cn+k,a(ˆxk,n+k)+xa1 kxa2 n+k. If bk−bn+k= 0fork=1,...,n,then{h1,...,h2n}has the L3/2-L3property by Theorem 5.5 with constant C(mink=1,...,n|bk−bn+k|)−1/3. This is the case in particular for {h1,...,h2n}as in (5.3). Hence, Theorems 3.1 and 1.8 are special cases of Theorems 5.8 and 5.16 below. We claim no originality for Theorem 5.5 that was applied in the previous example. It is an instance of much more general results available in the literature. We merely explain here how the statement follows from the L3/2-L3improving property of (i) the Radon transform and (ii) convolution with a measure on a parabola. Even though (i) involves integration over lines with different slopes, and (ii) concerns convolution with a fixed parabola, both operators fit in the same framework [43, p. 606]. Theorem 5.5 Let α, β, γ, δ, , κ ∈R.Ifβ= 0, then the operator S, defined by Sf(x,t)=R f(y,t+αy2+βxy +γx2+δx+y+κ)dy,f∈C∞ c(R2), satisfies Sf3|β|−1/3f3 2,f∈C∞ c(R2). (5.6) 123
K. Fässler, A. Pinamonti Proof We divide the proof in two cases: α=0andα= 0. In the first case, we apply the L3/2-L3improving property of the Radon transform [38](intheformofTheorem2.9). In the second case, we reduce matters to the L3/2-L3improving property of the convolution operator with a measure on a parabola [21,36,39]. First, if α=0, then, for f∈C∞ c(R2), we relate Sf to the operator Tfrom Theorem 2.9 as follows: Sf(x,t)=R f(y,t+[βx+]y+[γx2+δx+κ]) dy =Tf(βx+, t+γx2+δx+κ). Thus Sf3=R2|Tf(βx+, t+γx2+δx+κ)|3d(x,t)1 3 =|β|−1/3R2|Tf(ξ, τ)|3d(ξ, τ)1 3 =|β|−1/3Tf3, and hence Theorem 2.9 implies (5.6) in that case. If α= 0, we instead reduce matters to [36], or the more general [21, Theorem 1]. A special case of that theorem says that μα∗f3f3 2,f∈L3 2(R2), (5.7) with implicit constant independent of α,where μα∗f(x,t):= R f((x,t)−(y,αy2))|α|1/3dy, see also [39, Theorem 1]. To employ this result, we aim to relate Sf for f∈C∞ c(R2)to μα∗f. We apply elementary transformations to one of the expressions that appear in the definition of Sf, namely t+αy2+βxy +γx2+δx+y+κ =α'y+1 2β αx+ α(2+−α 4β αx+ α2+γx2+δx+κ+t. Hence, by the change-of-variables y→ η=−[y+1 2β αx+ α],weobtain Sf(x,t) =R fy,α'y+1 2β αx+ α(2+−α 4β αx+ α2+γx2+δx+κ+tdy =R f−1 2β αx+ α−η, −α 4β αx+ α2+γx2+δx+κ+t−(−α)η2dη =|α|−1/3μ−α∗f((x,t)), with (x,t):= −1 2β αx+ α,−α 4β αx+ α2+γx2+δx+κ+t. 123
Loomis–Whitney inequalities in Heisenberg... Since |det D(x,t)|=|β|| 2α|−1, we find that Sf3=|α|−1/3(μ−α∗f)◦3=|α|−1/3|β|−1/3|2α|1/3μ−α∗f3. Thus (5.6) in the case α= 0 follows from (5.7). We next prove a generalization of Theorem 3.1 that applies in particular to mappings ρ1,...,ρ 2nas in (5.1)forh1,...,h2nas in Example 5.4. Theorem 5.8 Let n ∈N. Assume that {h1,...,h2n}is a family of C1functions on R2nwith the L3/2-L3property and define ρj:R2n+1→R2n,ρ j(x,t)=ˆxj,t+hj(x),j=1,...,2n. Then, for all nonnegative Lebesgue measurable functions f1,..., f2non R2n,wehave R2n+1 2n j=1 fj(ρj(p)) dp fk2n+1 2fn+k2n+1 2 n j=1 j=kfj2n+1fn+j2n+1, k∈{1,...,n},(5.9) with an implicit constant that may depend on n and the boundedness constant C associated to the family {h1,...,h2n}.Ifn=1,then(5.9)reads R3 f1(ρ1(p)) f2(ρ2(p)) dp f13 2f23 2. The statement can be deduced by following the proof of Theorem 3.1 almost verbatim. We decided to give the argument for Theorem 3.1 first in Sect. 3since it is a bit easier to read and helps motivate the more general discussion in the present section. Below we merely explain how to adapt the proof of Theorem 3.1 to establish Theorem 5.8. Proof It suffices to verify the claim for nonnegative, smooth, and compactly supported functions f1,..., f2n. The case n=1 follows directly from the L3/2-L3property of {h1,h2}in Definition 5.2, and a simple change-of-variables argument, observing that R3 f1(ρ1(p)) f2(ρ2(p))dp =R3 f1(x2,t+h1(x1,x2)) f2(x1,t+h2(x1,x2)) d(x1,x2,t) =R2 f2(x1,τ)R f1(x2,τ +h1(x1,x2)−h2(x1,x2)) dx2d(x1,τ) =R2 f2(x1,τ)T1f1(x1,τ)d(x1,τ) ≤T1f13f23 2≤Cf13 2f23 2, for nonnegative f1,f2∈C∞ c(R2). Suppose next that the statement of Theorem 5.8 has already been established for all natural numbers up to n−1. We will argue that it holds also for the integer n. To this end, we fix an 123
K. Fässler, A. Pinamonti arbitrary family {h1,...,h2n}of C1functions R2n→Rwith the L3/2-L3property. Given nonnegative C∞ cfunctions f1,..., f2n, we aim to show the ninequalities stated in (5.9), and by symmetry it suffices to discuss this for k=1. By the same argument as in the proof of Theorem 3.1, but now using the transformation t→ t+h2n(x)=τ,wefindthat I:=R2nR 2n j=1 fj(ρj(x,t)) dt dx (5.10) =R2n f2n(ˆx2n,τ) ×⎡ ⎢ ⎢ ⎣R fn(ˆxn,τ +hn(x)−h2n(x)) 2n j=1 j=n,2n fj(ρj(x,τ −h2n(x))) dx2n⎤ ⎥ ⎥ ⎦d(ˆx2n,t). Applying Hölder’s inequality, we can split off the factor with f2n(which no longer depends on x2n) and we obtain I≤f2n2n+1Jwith J:= ⎡ ⎢ ⎢ ⎢ ⎣R2n⎛ ⎜ ⎜ ⎝R fn(ˆxn,τ +hn(x)−h2n(x)) 2n j=1 j=n,2n fj(ρj(x,τ −h2n(x))) dx2n⎞ ⎟ ⎟ ⎠ 2n+1 2n d(ˆx2n,t)⎤ ⎥ ⎥ ⎥ ⎦ 2n 2n+1 . The remaining task is to show that Jn,Cf12n+1 2fn+12n+1 2fn2n+1 n−1 j=2fj2n+1fn+j2n+1,(5.11) and this is done as in the proof of Theorem 3.1, but using the transformation τ→ t= τ+hn(x)−h2n(x). Then, as in the proof of Theorem 3.1, we find that in order to prove (5.11), it suffices to show that Jn,Cf12n+1 2fn+12n+1 2 n−1 j=2fj2n+1fn+j2n+1,(5.12) where J:= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝R ⎡ ⎢ ⎢ ⎢ ⎣R2n−1⎛ ⎜ ⎜ ⎝R 2n j=1 j=n,2n fj(ρj(x,t−hn(x))) 2n+1 2ndxn⎞ ⎟ ⎟ ⎠ 2n 2n−1 d(ˆxn,2n,t)⎤ ⎥ ⎥ ⎥ ⎦ 2n−1 2n dx2n⎞ ⎟ ⎟ ⎟ ⎟ ⎠ 2n 2n+1 . Applying Minkowski’s integral inequality inside the square brackets, then Fubini’s theorem and the transformation t→ τ=t−hn(x)yields J≤⎛ ⎜ ⎜ ⎜ ⎝R2⎡ ⎢ ⎢ ⎣R2n−1 2n j=1 j=n,2n fj(ρj(x,τ))2n+1 2n−1d(ˆxn,2n,τ) ⎤ ⎥ ⎥ ⎦ 2n−1 2n d(xn,x2n)⎞ ⎟ ⎟ ⎟ ⎠ 2n 2n+1 .(5.13) We recall that fj(ρj(x,τ))=fj(ˆxj,τ +hj(x)). (5.14) 123