Bergman kernels for Paley-Wiener spaces and Nazarov's proof of the Bourgain-Milman theorem
aa r X i v : . [ m a t h . C V ] A ug BERGMAN KERNELS FOR PALEY-WIENER SPACES AND NAZAROV’S PROOFOF THE BOURGAIN-MILMAN THEOREM.
BO BERNDTSSONA
BSTRACT . We give a general inequality for Bergman kernels of Bergman spaces defined byconvex weights in C n . We also discuss how this can be used in Nazarov’s proof of the Bourgain-Milman theorem, as a substitute for Hörmander’s estimates for the ¯ ∂ -equation.
1. I
NTRODUCTION If φ is a plurisubharmonic function in a domain D in C n , the Bergman space A ( D, e − φ ) is theHilbert space of holomorphic functions in D that are square integrable against the weight e − φ ; A = A ( D, e − φ ) = { f ∈ H ( D ); Z D | f | e − φ dλ < ∞} . The (diagonal) Bergman kernel for A is the function B ( z ) := sup f ∈ A | f ( z ) | / k f k . In this note we are mainly interest in the case when φ is convex and D = C n . However, wedo allow our convex functions to attain the value + ∞ , and in that case we are in effect onlyintegrating over the set where φ is finite, and we only require f to be holomorphic in the interiorof that set. With this understanding we omit the reference to the domain in the Bergman space,and write A = A ( e − φ ) . Our main result is as follows. Theorem 1.1.
Let φ and φ be convex functions on R n and let B be the Bergman kernel forthe space A ( e − φ ( x ) − φ ( x ) ) . Let also B ′ be the Bergman kernel for the space A ( e − φ ( x ) − φ ( y ) ) .Then, if φ and φ are symmetric ( φ j ( − x ) = φ j ( x ) ), B ′ (0) ≤ C n B (0) , where log C = 1 π Z − − log s s ds. If φ = φ = φ , we may take C = 2 if φ is homogenous of order 2, and C = 1 . if φ ishomogenous of order 1. The point of the theorem is that the spaces A := A ( e − φ ( x ) − φ ( x ) ) and A := A ( e − φ ( x ) − φ ( y ) ) ,are rather different and it is not a priori clear that their Bergman kernels should be in any wayrelated. While A is defined by a weight function that only depends on x , the weight in the definition of A decreases in all directions. Thus, e. g., the function f = 1 lies in A but not in A . Using that function in the definition of the Bergman kernel we get immediately B ′ (0) ≥ ( Z e − φ Z e − φ ) − , and the theorem implies that the same estimate holds for B (0) , up to a constant: Corollary 1.2.
With the notation of the theorem (1.1) B (0) ≥ C − n ( Z e − φ Z e − φ ) − Our principal motivation (or inspiration) for the theorem comes from Nazarov’s proof ([10])of the Bourgain-Milman theorem, [6]. Nazarov’s idea was to use Bergman kernels for Paley-Wiener spaces (see section 3) to prove inequalities for volumes of convex bodies. His main resultis basically what we have formulated as a corollary, in the special case when φ = φ = I K , theindicator of a symmetric convex body (defined as zero on K and infinity outside). The inequality(1.1) then becomes(1.2) B (0) ≥ C − n | K | − , where | K | means the volume of K . He then coupled this estimate with an estimate from above(1.3) B (0) ≤ n ! C n | K ◦ || K | − , where K ◦ is the polar body of K . The result is that | K || K ◦ | ≥ C n /n ! , with C an absolute constant. This is the Bourgain-Milman inequality. The famous Mahlerconjecture says that C can be taken equal to . This is so far open. The best result on the valueof the constant so far, C = π , is due to Kuperberg, [8].Nazarov used Hörmander’s L -estimates for the ¯ ∂ -equation to prove (1.2). Here we will usea different argument based on plurisubharmonic variation of Bergman kernels from [2]. Theinequality (1.3) is also non trivial, but much easier than (1.2), and completely elementary. Wewill give the corresponding inequality for convex functions in section 4 (this is based on themaster’s thesis of Hultgren, [7]).In the next section we will give the proof of Theorem 1.1. In section 3 we give some general-ities on Bergman and Paley-Wiener spaces, and in section 4 we discuss estimates from above ofBergman kernels ( in both these sections, we again largely follow [7]). In the last section we willdiscuss the values of the constant one can obtain by this method – they are somewhat worse thanKuperberg’s estimate, but just barely better than Nazarov’s.It should be said that the only new result in this note is Theorem 1.1. The rest of the material isat least essentially known, but we include it in an attempt to give an easy introduction to this veryinteresting circle of ideas. In a companion paper, [1], we also give a variation of Kuperberg’sproof, using different ideas from complex analysis. We also refer to [3] for another analysis ofNazarov’s proof, from a different angle.Finally I would like to thank Bo’az Klartag and Yanir Rubinstein for very stimulating discus-sions on these matters.
2. A
N ESTIMATE FOR B ERGMAN KERNELS
We are looking at Bergman spaces on C n defined by weight functions e − φ ( x ) where φ is convexon R n . We may assume that φ (0) = 0 and also assume that φ ( x ) = φ ( − x ) . The main technicaldifficulty in Nazarov’s paper is to prove that for φ = I K , the Bergman kernel at the origin isbounded from below by a constant depending on n times | K | − . This is where Hörmander’stheorem comes in.Here we follow another approach to prove such things in a somewhat more general setting. Let φ ( x ) and φ ( x ) be two convex functions on R n satisfying the hypotheses above. We considerthe Bergman space A ( e − φ ( x ) − φ ( x ) ) of holomorphic functions satisfying Z | f ( x + iy ) | e − φ ( x ) − φ ( x ) dxdy < ∞ . Let ζ ∈ C and define k f k ζ = Z C n | f | e − φ ( ζz ) − φ ( z ) dλ ( z ) . Here we are using the notation φ j ( z ) = φ j (Re z ) . Denote by B ζ the Bergman kernel for thisnorm. When ζ = i , the norm is k f k i = Z C n | f | e − φ ( − y ) − φ ( x ) dλ. The main point is that the Bergman kernel for this norm is fairly easy to estimate from belowsince f = 1 is a contender in the definition of the Bergman kernel. The theorem says that boundsfrom below of B i give bounds from below of B which is the Bergman kernel for the normdefined by φ + φ . Theorem 2.1.
For general convex functions φ , φ , symmetric and vanishing at the origin, wehave B i (0) ≤ C n B (0) , with log C = 1 π Z − − log s s ds. If φ = φ = φ , we may take C = 2 if φ is homogenous of order 2, and C = 1 . if φ ishomogenous of order 1. The proof depends on two things: First, log B ζ (0) =: b ( ζ ) is subharmonic in C . This is a directconsequence of the main theorem in [2] on plurisubharmonic variation of Bergman kernels since φ ( ζ z ) + φ ( z ) is plurisubharmonic in ( ζ , z ) . Second, it satisfies an estimate b ( ζ ) ≤ C + n log | ζ | . The second fact comes from changing variables k f k ζ = | ζ | − n Z C n | f ( z/ζ ) | e − φ ( z ) − φ ( z/ζ ) dλ ( z ) . This gives that B ζ (0) = | ζ | n ˇ B /ζ (0) , where ˇ B means that we have changed the roles of φ and φ . (It does not matter that f ( z ) changesto f ( z/ζ ) since the origin where we study the Bergman kernel remains fixed.) Then we note that ˇ B /ζ (0) is bounded as ζ → ∞ ; it tends to the Bergman kernel for the weight e − φ .This means that we can apply the Poisson representation in the upper half plane to the function b ( ζ ) − n log | ζ | , which is bounded near infinity. It is not bounded near the origin, where it has alogarithmic pole, but it is at any rate integrable on the real axis. Therefore we have that b ( i ) = log B i (0) ≤ π Z ∞−∞ b ( s ) − n log s s ds. What remains is to estimate b ( s ) for s ∈ R , and by symmetry we need only worry about s > .If s ≤ , we have(2.1) k f k s = Z | f | e − φ ( sx ) − φ ( x ) dλ ( z ) ≥ k f k , since φ ( sx ) is increasing in s . Hence b ( s ) ≤ b (1) .If s ≥ we have(2.2) k f k s = s − n Z | f | e − φ ( x ) − φ ( x/s ) dλ ( z ) ≥ s − n k f k , for the same reason. Hence b ( s ) ≤ b (1) + n log s when s ≥ . Notice that these estimates of b ( s ) are actually sharp if φ = φ is the indicator function of a convex body.Putting these estimates together we get b ( i ) ≤ π Z − b (1) − n log s s ds + 1 π Z | s | > b (1)1 + s ds. This completes the proof of the first part of the theorem.For the second part, when φ = φ = φ we have, when φ is homogenous of order 2, that k f k s = Z | f | e − φ ( sx ) − φ ( x ) dλ ( z ) = Z | f | e − φ ( √ ( s +1) / x ) dλ ( z ) == ( 21 + s ) n Z | f ( z/ p ( s + 1) / | e − φ ( x ) dλ ( z ) . This implies that B s (0) = ( s + 12 ) n B (0) , where B (0) is now the Bergman kernel for the weight function e − φ . Hence, using the Poissonintegral representation again, b ( i ) = log B i (0) ≤ n π Z ∞−∞ log(1 + s ) − log 21 + s ds + b (1) . (The term log s gives zero contribution to the Poisson integral.) We now use that π Z ∞−∞ log(1 + s )1 + s ds = 2 log 2 (it is a standard residue integral). Hence b ( i ) ≤ n log 2 + b (1) , which is what we claim.Finally, if φ = φ = φ where φ is homogenous of order 1, we write φ ( sx ) + φ ( x ) =2 φ ( x ( | s | + 1) / . The same argument as in the 2-homogenous case gives log C = 4 π Z ∞ log(1 + s )1 + s ds − . By numerical calculation (e.g. Wolfram alpha) this gives C slightly less than . . (cid:3) It is interesting to compare the second part of the theorem to the special case when φ ( x ) = | x | / . Then B i (0) = B (0) . To see this, note that x = ( x + y ) / z / Thus, if we multiply f by e z / , we change the second norm to the first. Hence the Bergmankernels are equal since e z / = 1 at the origin. I don’t know if the factor n in the theorem canbe omitted in general.Note also that when φ is x / , then(2.3) b ( ζ ) = n (log(1 + | ζ | ) − log 2) + b (1) , not only for ζ real, but for all ζ . This follows by a similar argument: Writing ζ = ξ + iη , | Re ( ζ z ) | = ξ | x | + η | y | − ξηx · y, which is equal to | ζ | | x | plus the real part of a holomorphic function ( of z ) vanishing at theorigin. Hence, the same argument as above gives 1.3. (Moreover, the same formula holds for anyquadratic form φ = x t A t Ax , where A is a real n × n matrix; this follows by changing variables w = Az .)So, when φ ( x ) = | x | / (or a general quadratic form), the estimate in the theorem is off by afactor n . We loose this factor in the proof when we use the Poisson representation formula. ThePoisson representation is an identity for harmonic functions, but for subharmonic functions weloose the Green potential part. For φ a quadratic form, the discussion above shows that the termwe loose is the Green potential of the Fubini-Study form i∂ ¯ ∂ log(1 + | ζ | ) .3. B ERGMAN SPACES , P
ALEY -W IENER SPACES AND LOGARITHMIC L APLACETRANSFORMS .If K is a convex body in R n , the Paley-Wiener space associated to K , P W ( K ) is the space ofall entire functions in C n of the form f ( z ) = Z K e t · z ˜ f ( t ) dt, where ˜ f lies in L ( K ) . We define the norm of f to be L -norm of ˜ f . Then P W ( K ) becomes aHilbert space of entire functions. Any f in the space satisfies an estimate | f ( x + iy ) | ≤ e h K ( x ) k f k p | K | , where h k ( x ) = sup t ∈ K t · x, is the support function of K and | K | denotes the volume of K . Moreover, the restriction of f tothe i R n lies in L . The Paley-Wiener theorem asserts that conversely, any such function f lies inthe Paley-Wiener space.We will consider a generalization of Paley-Wiener space, associated to convex functions on R n instead of convex bodies. If ψ is a convex function on R n , we denote by P W ( e ψ ) the spaceof all holomorphic functions of the form(3.1) f ( z ) = Z R n e t · z ˜ f ( t ) dt, where k f k P W := Z R n | ˜ f | e ψ dt < ∞ . We allow ψ to attain the value + ∞ , and the L -condition should then be interpreted so that ˜ f vanishes where ψ = ∞ . Thus the classical Paley-Wiener spaces correspond to ψ = I K ; theconvex indicator of K that is zero on K and infinity on the complement of K .Another particular case of generalized Paley-Wiener spaces are Bergman spaces, where theweight depends only on the real part of z . Let φ = φ ( x ) be a convex function on R n , andconsider the Bergman space, A ( e − φ ( x ) ) , associated to φ , i.e. the space of holomorphic functionssuch that(3.2) k f k A := Z | f ( x + iy | e − φ ( x ) dxdy < ∞ . Again, we allow φ to attain the value + ∞ . We then only integrate over the tube domain where φ is finite, and f is only required to be holomorphic there. Proposition 3.1 below says thatsuch spaces are Paley-Wiener spaces, associated to a weight function ψ that is the ’logarithmicLaplace transform’ of φ (see [4] and [5] for early uses of this transform). Definition 1.
Let φ be a convex function on R n . Then the logarithmic Laplace transform of φ isthe function ˜ φ defined by e ˜ φ ( t ) = Z e xt − φ ( x ) dx. Logarithmic Laplace transforms are clearly convex, but it is also clear that a general convexfunction can not be written like this. For one thing, they are always real analytic in the interiorof the set where they are finite, but it is also easy to see that e.g. the indicator of a convexbody is not a logarithmic Laplace transform. One can verify that the sum of two logarithmicLaplace transforms is again a logarithmic Laplace transform( of the convolution of e − φ and e − φ ; Prekopa’s theorem implies that the convolution is log-concave). Thus the set of logarithmic Laplace transforms is an additive semigroup and φ is again in the set, but (1 /
2) ˜ φ is in generalnot. Proposition 3.1.
The spaces
P W ( e ˜ φ ) and A ( e − φ ) are equal and k f k A = (2 π ) n k f k P W . Thus we see that weighted Bergman spaces where the weight only depends on x = Re z arePaley-Wiener spaces of a special kind, namely defined by weight functions ψ that are logarithmicLaplace transforms.For the proof, we first assume that f lies in P W ( e ˜ φ ) . Hence, f is the Fourier-Laplace trans-form (3.1) of a function ˜ f , and Z | ˜ f | e ˜ φ dt < ∞ . It follows that e t · x ˜ f lies in L ( R n ) for any x in the interior of the set where φ ( x ) < ∞ .Let f x ( y ) = f ( x + iy ) . Then Parseval’s formula gives Z | f x | dy = (2 π ) n Z e t · x | ˜ f | dt. Multiplying with e − φ ( x ) and integrating with respect to x we get Z | f ( x + iy ) | e − φ ( x ) dxdy = (2 π ) n Z | ˜ f | e ˜ φ dt. Thus f lies in A ( e − φ ( x ) ) and the norm in A coincides with the norm in P W ( e ˜ φ ) multiplied by (2 π ) n .Hence, the Paley-Wiener space with weight ˜ φ is at least isometrically embedded in A ( e − φ ( x ) ) ,and it remains only to show that it is dense.For this we take f in A and note that f x lies in L for almost all x in the interior of the setwhere φ ( x ) < ∞ . Take one such x . By Fourier inversion we can write f x ( y ) = Z e t · ( x + iy ) ˜ f ( t ) dt, with e t · x ˜ f in L . Now assume that ˜ f has compact support. Then f ( z ) = Z e t · z ˜ f ( t ) dt, since the right hand side is an entire function and it agrees with f where Re z = x . By Parseval’sformula again, Z | f ( x + iy ) | dy = (2 π ) n Z e t · x | ˜ f ( t ) | dt, and we can multiply by e − φ ( x ) and integrate with respect to x to conclude that f lies in P W ( e ˜ φ ) .Thus, to prove that the Paley-Wiener space is dense in A it suffices to prove that the space of f in A such that f x has compactly supported Fourier transform for some x , is dense. For this, let α be a smooth, non-negative function with compactly supported Fourier transform and totalintegral equal to 1. Put h ( z ) = Z f ( z + is ) α ( s ) ds. Then h x has compactly supported Fourier transform, and since | h ( z ) | ≤ Z | f ( z + is ) | α ( s ) ds,h lies still in A . Hence any such h lies in the Paley-Wiener space. Finally, by scaling α appropri-ately we get an approximate identity, and therefore a sequence of functions in the Paley-Wienerspace that converges to f . This completes the proof.We next give an explicit formula for the Bergman kernel of P W ( e ψ ) . Proposition 3.2.
The Bergman kernel B ( z ) for P W ( e ψ ) is B ( z ) = Z e t · x − ψ ( t ) dt. In particular , the Bergman kernel for A ( e − φ ) is (2 π ) − n Z e t · x − ˜ φ ( t ) dt. Proof.
By the definition of the Bergman kernel we have B ( x + iy ) = sup ˜ f | Z e t · ( x + iy ) ˜ f ( t ) dt | , where the supremum is taken over all ˜ f with Z | ˜ f | e ψ dt ≤ . The first part of the proposition follows immediately from this and Cauchy’s inequality. Thesecond part follows from the first part and formula (3.4). (cid:3)
Remark . There is an intriguing symmetry between Propositions 3.1 and 3.2 in that the loga-rithmic Laplace transform appears in both places, for rather different reasons. It would be niceto have a better explanation of this than just brute computation.Taking ψ = I K , P W ( e − ψ ) = P W ( K ) and the proposition shows that the Bergman kernelat the origin for the classical Paley-Wiener space P W ( K ) is | K | . This was the starting pointfor Nazarov. However, one cannot work directly in P W ( K ) , since it is difficult to constructfunctions in that space, by methods like Hörmander’s L -estimates. Therefore Nazarov worksinstead with ψ = ˜ φ , where φ is an indicator function. This is a Bergman space, and Hörmander-type methods can be applied. He uses a variant of Proposition 3.2 for the special case when φ = I K is the indicator of a convex body. We refer to [10] for a discussion of the origins of thisformula.
4. E
STIMATES FROM ABOVE OF THE B ERGMAN KERNEL
Following Nazarov, we next give an estimate from above of the Bergman kernel, with thedifference that we work with convex functions instead of convex bodies (see also [7]). In thenext theorem we take the weight to be φ ( x ) instead of φ ( x ) . This fits better with Theorem 1.1,and is also more convenient in the estimate from above. Theorem 4.1.
Let B be the (diagonal) Bergman kernel for the space A ( e − φ ( x ) ) . Assume φ isconvex and symmetric around the origin. Then (4.1) B (0) ≤ π − n R e − φ ∗ R e − φ , where φ ∗ is the Legendre transform of φ .Proof. Let
Φ = 2 φ . By Proposition 3.1(4.2) B (0) = (2 π ) − n Z e − ˜Φ , where e ˜Φ( t ) = Z e t · x − Φ( x ) dx. Changing variables we get e ˜Φ( t ) = 2 − n Z e t · x − Φ( x/ dx. Let x = y + ξ where y is constant and ξ is a new variable in the integral. By convexity, Φ(( y + ξ ) / ≤ (Φ( y ) + Φ( ξ )) / φ ( x ) + φ ( ξ ) , so e ˜Φ( t ) ≥ − n e t · y − φ ( y ) Z e t · ξ − φ ( ξ ) dξ. Since φ is symmetric, the gradient with respect to t of the integral in the right hans side vanishesfor t = 0 , so this is a minimum point for the function defined by the integral: Z e t · ξ − φ ( ξ ) dξ ≥ Z e − φ ( ξ ) dξ. Taking the supremum over all y we get e ˜Φ( t ) ≥ − n e φ ∗ ( t ) Z e − φ dx. Inserting this in formula (4.2) we get (4.1). (cid:3)
We can now combine this with Corollary 1.2 (with φ = φ = φ ) : Theorem 4.2. If φ is a symmetric convex function C − n π n ≤ Z e − φ Z e − φ ∗ where C is given in Theorem 1.1. If φ is homogenous of order 2, we can take C = 2 . This estimate is not optimal. The theorem of Kuperberg, [8] suggests that one can remove thefactor C − n , see [1] for a proof of this. The function version of Mahler’s conjecture says that onecan take the left hand side equal to n .5. C ONVEX BODIES AND NUMERICAL VALUES .In this section we translate Theorem 4.2 to an estimate for volumes of convex bodies andcompare the result obtained to earlier results.Take φ ∗ = I K , the indicator function of a symmetric convex body K . Then φ = h K , thesupport function of K . Hence Z e − φ ( t ) = Z R n dt Z ∞ h K ( t ) e − s ds = Z ∞ e − s Z h K ( t )