Bellman function approach to the sharp constants in uniform convexity

Transcription

1 Adv. Calc. Var. 08; (): 89 9 Research Article Paata Ivanisvili* Bellman function approach to the sharp constants in uniform convexity DOI: 0.55/acv Received February 9 06; revised May 5 06; accepted July 5 06 Abstract: We illustrate a Bellman function technique in finding the modulus of uniform convexity of L p spaces. Keywords: Uniform convexity Bellman function concave envelopes MSC 00: 8A0 6A06 5A0 5A40 4B0 4B5 47A0 Communicated by: Frank Duzaar Uniform convexity Let I be an interval of the real line. For an integrable function f over I we set f I := f(s) ds and f p := f p /p I. We recall the definition of uniform convexity of a normed space (X ) (see []). Definition ([]). The space X is uniformly convex if for any ε > 0 there exists δ > 0 such that if x = y = and x y ε then x+y δ. The modulus of convexity of the normed space X is defined as follows: δ X (ε) = inf( f + g ) : f = g = f g ε}. We notice that (X ) space is uniformly convex if and only if for any ε > 0 we have δ X (ε) > 0. Hanner (see []) gave an elegant proof of finding the constant δ L p(ε) > 0 for p ( ) in 955. He proved two necessary inequalities (further called Hanner s inequalities) in order to obtain the constant δ L p(ε). Namely f + g p p + f g p p ( f p + g p ) p + f p g p p p [ ] (.) and inequality (.) is reversed if p. Hanner mentions in his note [] that his proof is a reconstruction of some Beurling s ideas given at a seminar in Upsala in 945. In [] the non-commutative case of Hanner s inequalities was investigated. Namely Hanner s inequality holds for p [ /4] [4 ) and the case p (/4 4) (where p = ) was left open. In this paper we present a general systematic approach to some class of extremal problems by using a Bellman function technique where absolutely no background is required only elementary calculus. As an application we illustrate this approach in finding the constant δ L p(ε). We also show that the Bellman function (.) which arises naturally is a minimal concave function with the given boundary condition (.). *Corresponding author: Paata Ivanisvili: Department of Mathematics Kent State University Kent OHIO 4440 USA pivanisv@kent.edu Download Date /4/8 6: AM

2 90 P. Ivanisvili Bellman function approach to the sharp constants in uniform convexity Minimal concave functions: An abstract theorem Let Ω R n and let m and H be some arbitrary maps such that m : Ω R k and H : Ω R. Let Ω(I) denote the class of piecewise constant vector-valued functions φ : I Ω which take only finite number of values. Let conv(ω) denote the convex hull of the set Ω. We define the Bellman function as follows: B(x) = sup φ Ω(I) H(φ(t)) dt : Theorem. The following properties hold: () B is defined on the convex set conv[m(ω)] () B(m(y)) H(y) for all y Ω () B is concave function (4) B is minimal among those who satisfy properties () () and (). m(φ(t)) dt = x}. Proof. Fist we show property (). Let Dom B denote the domain where B is defined. Since m(φ) m(ω) we have m(φ) I conv[m(ω)]. Therefore Dom B conv[m(ω)]. Now we show the opposite inclusion. Carathéodory s theorem implies that for any x conv[m(ω)] we have x = n+ j= a jx j where a j 0 n+ j= a j = and x j m(ω). Let the points y j Ω be such that m(y j ) = x j. We choose φ so that t I : φ(t) = y j } = a j I. Then φ Ω(I). Hence m(φ) I = m(φ(t)) dt = n+ j= I t:φ(t)=y j } m(φ(t)) dt = To verify the second property take φ 0 (t) = y t I. Then m(φ 0 ) I = m(y). Thus B(m(y)) = For the third property it is enough to show that n+ j= sup H(φ) I H(φ 0 ) I = H(y). φ Ω(I): m(φ) I =m(y) B(θx + ( θ)y) θb(x) + ( θ)b(y) for all x y conv[m(ω)] and θ [0 ]. There exist functions φ ψ Ω(I) such that I x j I a j = x. m(φ) I = x m(ψ) I = y H(φ) I B(x) ε H(ψ) I > B(y) ε. We split the interval I into two disjoint subintervals I and I so that I = θ I. Let L j : I j I j = be linear bijections. We consider the concatenation as follows: Clearly η(t) Ω(I) m(η) I = θx + ( θ)y and η(t) = φ(l (t)) t I φ(l (t)) t I. B(θx + ( θ)y) H(η) I = θ H(φ) I + ( θ) H(ψ) I > θb(x) + ( θ)b(y) ε. Now we show property (4). Let G satisfy properties () () and (). Then Jensen s inequality implies that for any φ Ω(I) we have H(φ) I G(m(φ)) I G( m(φ) I ) = G(x). Before we proceed to the applications let us make the following observations. The requirement that φ(t) is piecewise constant and it takes finite number of values is excessive. We made this requirement in order to avoid issues with measurability. For example if H and m are arbitrary maps then the compositions m(φ) Download Date /4/8 6: AM

3 P. Ivanisvili Bellman function approach to the sharp constants in uniform convexity 9 and H(φ) may not be measurable and the integral averages H(φ) I and m(φ) I will not make sense. However if H m and φ are Borel measurable then the above argument does not change. We note that property () of Theorem namely B(m(y)) H(y) for all y Ω can be rewritten as B(x) sup y m (x) H(y) for all y m(ω). We define the obstacle by R(x) := sup y m (x) H(y). Then it is clear that B(x) = sup R(m(φ)) I : m(φ) I = x} φ Ω(I) and property () takes the form B(x) R(x) for all x m(ω). The Bellman function B does not depend on the choice of the interval I and B(x) = R(x) at the extreme points of the set conv[m(ω)]. This follows from the fact that a minimal concave function must coincide with its obstacle at the extreme points of its domain. If conv[m(ω)] = m(ω) and the obstacle R(x) is concave then B(x) = R(x) for all x m(ω). In the next section we apply Theorem to find δ L p(ε). The Bellman function in uniform convexity. The domain and the boundary condition The definition of the modulus of uniform convexity tells us to consider the following function: B(x) = sup θf + ( θ)g p I ( f p g p f g p ) I = x} (.) fg where θ = / and the supremum is taken over real valued Borel measurable functions f g L p (I). Then it is clear that δ L p(ε) = sup p x ε p(b( x )) /p. Theorem implies that if we set Ω = R φ(t) = (f(t) g(t)) m(x y) = ( x p y p x y p ) and H(x y) = θx + ( θ)y p then B is the minimal concave function on the domain conv[m(ω)] such that B(m(x y)) H(x y). If we set m(φ) I = x = (x x x ) then all variables x x x are nonnegative. We notice that Λ := conv[m(ω)] = x x x 0 : x /p + x /p x /p x /p + x /p x /p x /p + x /p x /p }. Λ is the convex cone and Λ = m(ω). Minkowski s inequality implies that whenever x m(ω) we must have f = λg for an appropriate λ. This allows us to find the boundary data for B B(x x x ) = θx/p (θx /p (x /p ( θ)x /p p x /p + x /p = x /p + x /p )p x /p + x /p = x /p + ( θ)x /p )p x /p + x /p = x /p (.) where x x and x are nonnegative numbers. Thus Theorem implies that B is a minimal concave function on Λ with the given boundary condition (.). So the questions becomes purely geometrical: given the convex domain Λ find the minimal concave function B in the convex domain Λ with the given boundary condition (.) on Λ. Then δ L p(ε) = sup (B( x )) /p. p x ε p We notice that the knowledge of B gives more information about the structure of uniform convexity and not just the value δ L p(ε). Remark. For the application one can avoid finding the exact value of B(x x x ). Indeed one can take an arbitrary concave function B on Λ which dominates B on Λ (the set where the obstacle H lives) and then by Theorem (namely property (4)) we have B B on Λ automatically. By the appropriate choice of B one may obtain the exact value of δ L p(ε). In the next subsection we will illustrate this idea. Download Date /4/8 6: AM

4 9 P. Ivanisvili Bellman function approach to the sharp constants in uniform convexity. Sharp constants in uniform convexity First we consider the case p. Let us consider the following function: B(x x x ) = x + x x p. (.) It is clear that B is concave in Λ. We notice that (B B) Λ 0 (see Lemma ). Therefore Theorem (4) implies that B B in Λ. Thus δ(ε) = sup (B( x )) /p p x ε p sup (B( x )) /p p x ε p = (B( ε p )) /p = ( εp p ) /p. If we show that B( ε p ) = B( ε p ) then this would imply that the estimate obtained above for δ L p(ε) is sharp. For the benefit of the reader we include the proof of this equality in Appendix A. Now we consider the case < p <. We set (g(s) f(s)) = ( s /p p (s /p /) p ) for s p and for all p ( ). Let s [ p ) be the solution of the equation ε p = s + g(s ). Consider the function B(x x x ) = x f(s ) + f (s ) + g (s ) [x + x ε p x ] (.4) which is concave in Λ. By Lemma we have (B B) Λ 0. Therefore Theorem (4) implies that B B in Λ. Thus δ(ε) = sup (B( x )) /p p x ε p sup (B( x )) /p p x ε p = (B( ε p )) /p = ε(f(s )) /p. We have used the fact that B( x ) is decreasing in x. Indeed the inequality B x ( x ) 0 follows from the inequality f(s)( + g (s)) f (s)(s + g(s)) 0 for all s p which can be seen by direct computation. The equality B( ε p ) = B( ε p ) (see Lemma ) implies that the estimate obtained above for δ L p(ε) is sharp. It is clear that the constant δ L p(ε) [0 ] is the solution of the equation ( δ(ε) + ε )p + δ(ε) ε p =. One can ask how did we find the functions B. These functions are tangent planes to the graph of the actual Bellman function B at the point ( ε p ). Unlike the actual Bellman function B which has the implicit expression the tangent planes B to the graph B have simple expression so it was easy to work with the tangent planes B rather than with the actual Bellman function B. The actual Bellman function B can be constructed by using the methods recently developed in [4 6]. Finally we should explain the appearance of the functions f(s) and g(s) for s p. Let us consider the section of the convex set Λ and the hyperplane x = in R. The result Λ (x x x ) : x = } is a convex set in the x x plane and it has a boundary. The boundary as a curve is symmetric along the line x = x which intersects it at the point ( p p ) and it can be identified with the union of the set of points (s g(s)) : s p )} (g(s) s) : s p )}. The function f(s) represents the value of B on the curve (s g(s)) i.e. it follows from (.) that B(s g(s) ) = B(g(s) s) = f(s) for s p. So the appearance of the curve (g(s) f(s)) for s p is natural and the point ( p p ) should be understood as the vertex of the curve (Λ (x x x ) : x = }) in R. Download Date /4/8 6: AM

5 P. Ivanisvili Bellman function approach to the sharp constants in uniform convexity 9 A Appendix Lemma. Let B be defined as in (.) for p and as in (.4) for < p <. Let B be defined as in (.) with the boundary conditions (.). Then (B B) Λ 0. Proof. Notice that (.) implies that B is -homogeneous function i.e. B(λx) = λb(x). It is clear that B is also -homogeneous as a linear function passing through the origin. It follows that without loss of generality we can assume x =. In this case boundary condition (.) can be rewritten as B(s g(s) ) = B(g(s) s ) = f(s) for all s p. It is enough to show that U(s) = B(s g(s) ) f(s) 0 for s p because the other cases can be covered by symmetry i.e. B(s g(s) ) = B(g(s) s ) and B(s g(s) ) = B(g(s) s ). The case p. Let V(s) = U(s p ). Note that V( ) = 0 and V (s) 0 for s. Note also that W(s) = V (s )p s p = (s ) p ( s )p is a concave function for s ( ) and it is nonnegative at the endpoints of this interval. Hence V (s) 0 for s. The case < p <. Note that U(s ) = 0 the function clear that U (s)( + g (s)) = f (s) +g (s) is decreasing and + g (s) > 0 for s p. It is f (s ) + g (s ) f (s) + g (s). Thus we see that U (s) < 0 for s [ p s ) U (s ) = 0 and U (s) > 0 for s (s ). Hence U(s) 0. Lemma. Let B and B be defined as above. Then B( ε p ) = B( ε p ) for all p ( ) and all > ε > 0. Proof. The -homogeneity of the functions B and B implies that it is enough to prove the equality B(ε p ε p ) = B(ε p ε p ). The case p. We show that B(s s ) = B(s s ) for all s p. Take an arbitrary s ( p ). Consider the points A = ( p p ) and D(s) = (s s /p p ). Clearly A D(s) Λ. Let L s (x) be a linear function such that L s (A) = B(A) = 0 and L s (D(s)) = B(D(s)) = ( / + s /p ) p. The concavity of B implies that B L on the chord [A D(s)] joining the points A and D(s). On the other hand one can easily see that for all a > 0 there exists sufficiently large s such that L s (x) B(x) < a for the points x belonging to the chord [A D(s)]. The continuity of B and B proves the assertion. The case < p <. Consider the points A = (s g(s ) ) and D = (g(s ) s ). Clearly A D Λ. Let L(x) be a linear function such that L(A) = B(A) = f(s ) and L(D) = B(D) = f(s ). The concavity of B implies that B(x) L(x) on the chord [A D] joining the points A and D. On the other hand one can easily see that B(x) = L(x) for the points x belonging to the chord [A D]. Thus the fact (ε p ε p ) [A D] which follows from the equality ε p = s + g(s ) finishes the proof. References [] K. Ball E. A. Carlen and E. H. Lieb Sharp uniform convexity and smoothness inequalities for trace norms Invent. Math. 5 (994) no [] J. A. Clarkson Uniformly convex spaces Trans. Amer. Math. Soc 40 (96) no [] O. Hanner On the uniform convexity of L p and l p Ark. Mat. (956) [4] P. Ivanisvili Inequality for Burkholder s martingale transform Anal. PDE 8 (05) no [5] P. Ivanisvili N. Osipov D. Stolyarov V. Vasyunin and P. Zatitskiy Bellman function for extremal problems in BMO Trans. Amer. Math. Soc. 68 (06) no [6] P. Ivanisvili D. Stolyarov V. Vasyunin and P. Zatitskiy Bellman function for extremal problems in BMO II: Evolution Mem. Amer. Math. Soc. to appear. Download Date /4/8 6: AM