An extension of Alexandrov s theorem on second derivatives of convex functions

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1 Avances in Mathematics 228 ( An extension of Alexanrov s theorem on secon erivatives of convex functions Joseph H.G. Fu 1 Department of Mathematics, University of Georgia, Athens, GA 362, USA Receive 17 August 29; accepte 1 July 211 Available online 2 July 211 Communicate by Erwin Lutwak Abstract If f is a function of n variables that is locally L 1 approximable by a sequence of smooth functions satisfying local L 1 bouns on the eterminants of the minors of the Hessian, then f amits a secon orer Taylor expansion almost everywhere. This extens a classical theorem of A.D. Alexanrov, covering the special case in which f is locally convex. 211 Elsevier Inc. All rights reserve. Keywors: Secon erivatives; Weak erivatives 1. Introuction Let U R n be open. We say that f : U R is twice ifferentiable at x if there is a quaratic polynomial Q x such that f(y Q x (y lim y x y x 2 =. (1 In the 193s A.D. Alexanrov [2] prove that if f : U R is a locally convex function on a omain U R n then f is twice ifferentiable almost everywhere. In the present article we exten this conclusion to a much larger class of functions. aress: fu@math.uga.eu. 1 Partially supporte by NSF grant DMS /$ see front matter 211 Elsevier Inc. All rights reserve. oi:1.116/j.aim

2 J.H.G. Fu / Avances in Mathematics 228 ( Theorem 1.1. Let U R n be open, an f 1,f 2,... C 2 (U converge in L 1 loc (U to a function f. Suppose that the absolute integrals of all minors of the Hessians of the f k are uniformly locally boune, i.e. K ( 2 et f k x i x j i I,j J C(K, k = 1, 2,... (2 whenever K U an I,J {1,...,n} have the same carinality. Then (1 the f k are locally uniformly boune, (2 f k f pointwise a.e. in U, (3 both lim sup k f k an lim inf k f k are twice ifferentiable at a.e. x U. Uner the hypotheses of the theorem we will say that f 1,f 2,... is a strong approximation of f. Strongly approximable functions belong to the class of Monge Ampère functions introuce by the present author in [6,7] an generalize by R.L. Jerrar in [11,12]. They are istinguishe by the existence of an integral current D(f in the cotangent bunle T U, representing graph of its ifferential. The key point is that D(f is etermine uniquely (if it exists by a short list of inevitable conitions. Remarks. 1. By the BV compactness theorem, the hypothesis of Theorem 1.1 implies local W 1,1 convergence of a subsequence. 2. If n = 1 then f is strongly approximable iff it is expressible as the ifference of two convex functions. If n = 2 then any ifference of convex functions is strongly approximable, but for n 2 there exist strongly approximable functions that are not ifferences of convex functions. Furthermore, there are pairs of strongly approximable functions whose sum is not strongly approximable (see below. If n 3 then it is not known whether the ifference of convex functions is strongly approximable, or even whether it is Monge Ampère. Alexanrov s theorem is a special case of Theorem 1.1: if f is convex then any locally uniform approximation by smooth convex functions (as may be obtaine, for example, by convolution with an approximate ientity is a strong approximation. But Theorem 1.1 is significantly stronger: even though a convex function f may fail to have first erivatives in the usual sense on a ense set, G. Alberti an L. Ambrosio [1] observe that such f amits a multiple-value ifferential everywhere, whose graph transforms into the graph of a Lipschitz function uner the linear change of variable (x, y (x + y,x y of T R n R n R n. From this point of view, Alexanrov s theorem appears as a consequence of Raemacher s theorem on the almost everywhere ifferentiability of Lipschitz functions. On the other han, there exist strongly approximable functions isplaying much wiler behavior. For example, they inclue the Sobolev space W 2,n loc (U of functions with secon istributional erivatives in L n loc (U, since Höler s inequality implies that convolution with an approximate ientity yiels a strong approximation. Meanwhile, Hutchinson an Meier [1] observe that if n 2 then f HM (x 1,...,x n := x 1 sin log log x 1 (3

3 226 J.H.G. Fu / Avances in Mathematics 228 ( belongs to W 2,n (U for sufficiently small neighborhoos U of, while f HM oscillates infinitely often between (±1,,..., + o(x as x along the x 1 axis. This implies first of all that f HM cannot be expresse as a ifference of two convex functions. Furthermore, by cutting off, translating, rotating an multiplying by suitable constants, it is easy to construct a W 2,n (R n - convergent sum whose ifferential has a graph that is ense in T R n. For n = 2 the function g(x,y := f HM (x, y + y is not Monge Ampère, an hence not strongly approximable. For the restriction of g to R 2 {} is Monge Ampère, an along the x 1 axis the fibers of the ifferential current D(g inclue the line segments with enpoints f HM ± (, 1 R 2. In view of the oscillation escribe above it follows that D(g oes not have finite mass above any neighborhoo of the origin. This example is easily extene to higher imensions. It is natural to conjecture that Theorem 1.1 applies to all Monge Ampère functions inee it seems plausible that every Monge Ampère function is strongly approximable. However, this is just one of many perplexing questions about Monge Ampère functions: for example, if n>2we o not even know whether such functions are necessarily continuous, or even locally boune. 2. Basic facts 2.1. Some measure theory Put B(x,r R n for the open ball of raius r about x, an ω n for the volume of B(, 1. Recall that if μ is a Raon measure on U R n then its ensity at x U is Θ(μ,x := lim r μ(b(x, r ω n r n, provie the limit exists. In fact the limit exists for a.e. x U with respect to Lebesgue measure, an efines a Lebesgue-integrable function of x, whose integrals yiel the absolutely continuous part of μ with respect to the Lebesgue ecomposition into absolutely continuous an singular parts (cf. [5, Theorem 3.22]. In particular, if μ is singular with respect to the Lebesgue measure then Θ(μ,x = for a.e. x R n Absolute Hessian eterminant measures Suppose that f 1,f 2,... f is a strong approximation. For =,...,n, an k = 1, 2,..., we efine the measures ν k, on U by ν k, (S := I,J {1,...,n}, I = J = S [ 2 ] et f k x i x j i I,j J. (4 Taking subsequences, we may assume that each sequence ν k,, k = 1, 2,..., converges weakly to a Raon measure ν, =,...,n. We will refer to any Raon measure ν ν as an absolute Hessian eterminant measure of egree for the strong approximation f 1,f 2,... f. When the approximation is unerstoo we will also refer to such ν as an absolute Hessian eterminant measure of egree for f. For quaratic polynomials Q : R n R, put Q to be the maximum of the absolute values of the coefficients. The following lemma is obvious.

4 J.H.G. Fu / Avances in Mathematics 228 ( Lemma 2.1. There are constants C n, with the following property. Let f L 1 loc (U be strongly approximable, an let ν 1,...,ν n be absolute Hessian eterminant measures for f arising from a strong approximation f 1,f 2,... f. Given any quaratic polynomial Q, the sequence f 1 + Q, f 2 + Q,...is then a strong approximation of f + Q, an the measures ν := C n, i+j= Q i ν j (5 are absolute Hessian eterminant measures of egrees = 1, 2,...,nfor this approximation. Remark. Since the number of minors of an n n matrix is ( n 2, an there are i= ( 2 i = ( 2 subminors of each such minor, the rather extravagant value Cn, =! ( n 2 ( 2 works An inequality from multivariable calculus The key fact that makes the main theorem work is the following elementary classical inequality about C 2 functions. Lemma 2.2. Let U R n be open, an F C 2 (U. IfV U is open then et D 2 F ( supv F sup ω V F n n. (6 iam V V Proof. This is a weakene form of Lemma 9.2 of [8]. 3. An extension of a special case of a theorem of Calerón an Zygmun We say that f : R U R amits a kth erivative in the L 1 sense at x if there exists a polynomial Q of egree k such that r n B(x,r f(y Q(y y = o ( r k (7 as r. The next proposition generalizes a result of Calerón an Zygmun [3]. The original result (or rather the very special case of it that we have in min states that a function with istributional secon erivatives in L 1 loc amits a secon erivative in the L1 sense a.e. By a straightforwar aaptation of the argument of [3] we prove that this conclusion is true if the istributional secon erivatives are only locally finite signe measures. Recall that the space BV loc (U of functions of locally boune variation consists of all locally integrable functions whose istributional graients are (vector measures (cf. [9]. Lemma 3.1. If g BV loc (U then g is ifferentiable in the L 1 sense at a.e. x U. Proof. This is an immeiate consequence of Theorem of [4].

5 2262 J.H.G. Fu / Avances in Mathematics 228 ( Proposition 3.2. If the istributional graient of f L 1 (U lies in BV loc (U, then f amits a secon erivative in the L 1 sense a.e. in U. Proof. By Lemma 3.1 it is enough to show that there exists an L 1 quaratic Taylor approximation for f at provie is a Lebesgue point of f an f is ifferentiable in the L 1 sense at. We may assume that f( = f( = D 2 f( =, where D 2 f( is the L 1 erivative of f at. Put G(ρ := B(,ρ f(x x n 1 x, F(ρ := B(,ρ f(x x. Then F, G are both absolutely continuous on [,, with G (ρ = F (ρ ρ n 1, an F(ρ= o(ρ n+1. Integrating by parts, it follows that G(ρ = o(ρ 2 as ρ. On the other han, B(,ρ f C C ρ ρ = Cρ n r n 1 r r n 1 r S n 1 v S n 1 v S n 1 v ρ r ρ Df (sv s = Cρ n G(ρ = o ( ρ n+2, Df (sv s Df (sv s which gives the result. 4. Proof of Theorem 1.1 For the rest of the paper we take as given the hypotheses of Theorem Proof of conclusion (1 The proof is a simplifie version of the proof of the other conclusions. Proposition 4.1. The f k are locally uniformly boune. Proof. Given x U an s>, put Q(x, s := [x 1 s,x 1 + s] [x n s,x n + s] (8

6 J.H.G. Fu / Avances in Mathematics 228 ( for the close cube of sie 2s centere at x. Letr > be small enough that Q(x, r U. We will show that for some constant C inepenent of k. For z (,r n, put sup f k C (9 Q(x, r 2 R(z = R(x,z := [x 1 z 1,x 1 + z 1 ] [x n z n,x n + z n ]. (1 Let F (z enote the set of -imensional faces of R(z. ForF F (z, put F for the - imensional affine space it generates. Consier the functions g k, : (,r n R given by g k, (z := g k, (z := G k := F F (z F F F (z F Q(x,r y F (z et D 2 ( f k F fk (y, et D 2( f k F, = 1,...,n, n g k,, k= 1, 2,... (11 = Then by Fubini s theorem an (11 g k, r (,r n I = Q(x,r (,r n g k, = [ 2 ] et f k x i x j i,j I C, f k C, Q(x,r = 1,...,n, for some constant C, inepenent of k, where the sum is over all subsets I {1,...,n} of the inicate carinality. By Fatou s lemma, lim inf G k lim inf G k (n + 1C. k k (,r n (,r n Taking a subsequence if necessary we may therefore fin z (,r n (, r 2 n such that g k, (z G k (z 2n 2 n 1 (n + 1Cr n =: C for all sufficiently large k.

7 2264 J.H.G. Fu / Avances in Mathematics 228 ( In particular f k (v <C for all of the vertices v F (z of the rectangle R(z. Applying Lemma 2.2 to the faces of R(z we fin that sup f k sup f k +ω 1 ( g k, z r sup f k +ω 1 ( C r, F F (z F F 1 (z F F 1 (z = 1,...,n. Proceeing by inuction on, the case = n yiels ( Proof of conclusions (2 an (3 The proof of Theorem 1.1 will be complete in the following proposition. Put ν := ν 1 + +ν n for the sum of the absolute Hessian eterminant measures of f, an ν = ν ac + ν s the ecomposition of ν into its absolutely continuous an singular parts. Let φ enote the ensity function of ν ac with respect to Lebesgue measure. We put also f := lim sup f k, k f := lim inf f k. k Proposition 4.2. Suppose x U, an that (1 Θ(ν s,x = ; (2 x is a Lebesgue point of f,of f an of φ; (3 f is twice ifferentiable in the L 1 sense at x. Then f(x = f (x, an both f, f are twice ifferentiable at x. Proof. We may assume that x =, an by Lemma 2.1 we may also assume that f( = f( = D 2 f( =, where D 2 f( is the L 1 secon erivative of f at. Let ɛ (, 1 be given, an take r > small enough that if r (,r then, referring to the efinition (8, ν s ( Q(, 2r + Q(,2r ( f ɛ 2 r 2 + φ(x φ( <(ɛr n. (12 Let x Q(,r. We will show that for large k f k (x <Mɛ 2 x 2 (13 where M epens only on φ( an the imension n. Put r := max( x 1,..., x n an h k, := I = J = [ 2 ] et f k x i x j i I,j J. (14

8 J.H.G. Fu / Avances in Mathematics 228 ( Then lim sup k Q(x,ɛr h k, ν ( Q(x, ɛr = ν s ( Q(x, ɛr + Q(x,ɛr ν s ( Q(, 2r + (2ɛr n φ( + φ Q(,2r φ(y φ( < ( n φ( (ɛr n (15 by (12. Recalling the efinitions of (1 ff., note that g k, (z F F (z F Q(x,ɛr et D 2 ( f k F, = 1,...,n for z (,rɛ n. Clearly (,ɛr n g k, (z = Q(x,ɛr fk (w <(ɛr n+2 for large values of k, by (12 an the local L 1 convergence f k f. Furthermore n (ɛr =1 (,ɛr n g k, (z z = n (ɛr =1 n =1 Q(x,ɛr (,ɛr n z h k, <(ɛr n( n φ(, for k large enough, by (15 an Fubini s theorem. Putting H k (z := 1 ɛ 2 r 2 g k,(z + F F (z F Q(x,ɛr = 1,...,n, n (ɛr g k, (z, =1 et D 2( f k F Fatou s lemma yiels lim inf H k(z z lim inf H k (z z k k (,ɛr n (,ɛr n <(ɛr n( n φ(.

9 2266 J.H.G. Fu / Avances in Mathematics 228 ( Therefore, taking subsequences as necessary, there exists z (,ɛ n such that H k ( z < n φ( =: C (16 for all sufficiently large k. For k large the values of f at the vertices v F (z of R(z satisfy f k (v <C(ɛr 2. Proceeing by inuction on the imension, we claim that for large k sup f k (y CA ɛ 2 r 2, y F F (z = 1,...,n, where C is the constant from (16 an A epens only on. To see this we observe that (16 yiels F et D 2( f k F <C(ɛr, = 1,...,n, (17 for F F (z an large k. Thus by Lemma 2.2, for each such face F whence ( sup F f k sup f k F (iam F ω 1 C(ɛr Cω 1 (ɛr2 2 sup F f k sup F f k +C 1 1 ω (ɛr 2 ( CA 1 + C 1 ω 1 (ɛr 2, as claime. Now (13 hols with M = CA n. A similar, easier argument shows that lim k f k ( = = f(. Acknowlegments I woul like to thank Bob Jerrar for stimulating conversations on the topics iscusse here, an also the Università i Trento for their hospitality uring part of this work. I am extremely grateful also to the referee, who provie a much shorter an more irect proof of the main theorem. In fact the new proof yiels a stronger statement than the original.

10 J.H.G. Fu / Avances in Mathematics 228 ( References [1] G. Alberti, L. Ambrosio, A geometrical approach to monotone functions in R n, Math. Z. 23 ( [2] A.D. Alexanrov, Almost everywhere existence of the secon ifferential of a convex function an some properties of convex surfaces connecte with it, Leningra Univ. Ann. (Math. Ser. 6 ( (in Russian. [3] A.P. Calerón, A. Zygmun, Local properties of elliptic partial ifferential equations, Stuia Math. 2 ( [4] L.C. Evans, R. Gariepy, Measure Theory an Fine Properties of Functions, CRC Press, Boca Raton, [5] G.B. Follan, Real Analysis, Wiley, New York, [6] J.H.G. Fu, Monge Ampère functions I, Iniana Univ. Math. J. 38 ( [7] J.H.G. Fu, Monge Ampère functions II, Iniana Univ. Math. J. 38 ( [8] D. Gilbarg, N. Turinger, Elliptic Partial Differential Equations of Secon Orer, Springer, New York, [9] E. Giusti, Minimal Surfaces an Functions of Boune Variation, Birkhäuser, Boston, [1] J.E. Hutchinson, M. Meier, A remark on the nonuniqueness of tangent cones, Proc. Amer. Math. Soc. 97 ( [11] R.L. Jerrar, Some remarks on Monge Ampère functions, in: Stanley Alama, Lia Bronsar, Peter J. Sternberg (Es., Singularities in PDE an the Calculus of Variations, in: CRM Proc. Lecture Notes, American Mathematical Society, 28. [12] R.L. Jerrar, Some rigiity results relate to Monge Ampère functions, Cana. J. Math. 62 (