Symmetrization of plurisubharmonic and convex functions

Theorem 2.1. Assume that D is a pseudoconvex domain in Cn+1 such that all its slices Dt are connected and S1-invariant. Assume also that the origin belongs to Dt when t lies in a domain U in C. Then log |Dt| is a superharmonic function of t in U.

Theorem 2.1 is a consequence of the main result in [2] which says that if Bt(z, z) is the Bergman kernel on the diagonal for the domain Dt, then log Bt(z, z) is plurisubharmonic in D. The hypotheses on Dt in the theorem imply that Bt(0, 0) = |Dt|−1 which gives the theorem. Theorem 2.1 can be seen as a complex variant of the (multiplicative form of) the Brunn-Minkowski inequality, which says that if D is instead convex in Rn+1, and the n-dimensional slices are defined in the same way then log |Dt|

Lemma 2.2. Let u be a smooth subharmonic function defined in an open set U in RN, and assume that u vanishes on the boundary of U. Let σ(t) := {x; u(x) < t}| for t < 0. Then σ is strictly increasing on the interval (min u, 0) and the Schwarz symmetrization of u, ˆu, equals g(|x|) where g(r) = σ−1(cNrN) where cN is the volume of the unit ball in RN, when cNrN > |Umin(u)|. When cNrN ≤|Umin(u)|, g(r) = min(u)

Theorem 2.3. Let Ωbe a balanced domain in Cn. Let φ be an S1-invariant plurisubharmonic function in Ω. Then ˆφ, the Schwarz symmetrization of φ is plurisubharmonic.

Theorem 2.1 can be applied and we conclude that log σ(Re τ) = log |Dτ| is a superharmonic function of τ. Since this function only depends on Re τ it is actually concave, and the proof is complete. □ The next theorem is the main result of this paper, and here we need to assume that Ωis a ball. See the remarks below for a discussion of the problem with considering more general domains.

Theorem 2.4. Let φ be plurisubharmonic in the unit ball, and assume that φ extends continuously to the closed ball with zero boundary values. Assume also that φ is S1-invariant, and let ˆφ be the Schwarz symmetrization of φ. Then E(ˆφ) ≤E(φ). In the proof of Theorem 2.1 we used the geometrically obvious fact that the inverse of an increasing concave function is convex. We will need a generalization of this that we state as a lemma.

Lemma 2.5. Let a(s, t) be a concave function of two real variables. Assume a is strictly increas- ing with respect to t, and let t = k(s, x) be the inverse of a with respect to the second variable for s fixed, so that a(s, k(s, x)) = x. Then k is convex as a function of both variables s and x.

Proposition 2.6. Let φt be a subgeodesic of S1-invariant plurisubharmonic functions. Then ˆφt is also a subgeodesic.

Lemma 2.4. □ We now first sketch the principle of the argument and fill in some details and change the set up a little bit afterwords. Consider the energy functionals along the two curves φt and ˆφt, E(φt) =: g(t) and E(ˆφt) = h(t). Since φ0 is already radial, g(0) = h(0), and we want to prove that g(1) ≥h(1). We know that g is affine and that h is concave, so this follows if we can prove that g′(0) = h′(0). But g′(0) = Z −˙φ0(ddcφ0)n,

Lemma 2.7. Let u be a smooth function such that D := {(w, z); u(z) −Re w < 0} is pseudoconvex. Then u is plurisubharmonic.

Proposition 2.8. If φ = f(uΩ) with f convex, the Monge-Ampere energy of φ equals Z Ω (−φ)(ddcφ)n = 2−n Z 0 −∞ (f ′)n+1(t)dt. In particular, the energy of φ is equal to the energy of ˆφ, the Schwarz symmetrization of φ. In the proof we use the next lemma.

Lemma 2.9. If φ = f(uΩ) with f convex Z uΩ<s (ddcφ)n = 2−nf ′(s)n.

Proposition 2.10. Let Ωbe a balanced domain in Cn and let uΩbe the uniquely determined logarithmically homogenous (plurisubharmonic) function that vanishes on the boundary of Ω. Assume uΩsatisfies the condition that (2.3) be constant on some, and therefore every , level surface of uΩ. Then Ωis an ellipsoid Ω= {z; X ajkzj¯zk < 1} for some positively definite matrix A = (ajk).

Theorem 2.11. Let Ωbe a strictly pseudoconvex balanced domain for which the symmetrization inequality EB(ˆφ) ≤EΩ(φ) holds for all S1-invariant plurisubharmonic φ that vanish on the boundary. Then Ωis an ellip- soid. 3. A SHARP MOSER-TRUDINGER INEQUALITY FOR S1-INVARIANT FUNCTIONS. Our results in the previous section, together with Moser’s inequality imply rather easily the next estimate.

Theorem 3.1. Let φ be a smooth S1-invariant plurisubharmonic function in the unit ball that vanishes on the boundary. Let E := E(φ). Then Z B enE−1/n(−φ)n+1)/n ≤C where C is an absolute constant. In the proof we may by our main result on symmetrization assume that φ(z) = f(log |z|) is a radial function. The main result of Moser, [5], is that if w is an increasing function on (−∞, 0) that vanishes when t goes to zero and satisfies Z 0 −∞ (−w′)n+1dt ≤1 then

Lemma 3.2. Let f be an increasing convex function on (−∞, 0] with f(0) = 0, and let φ(z) = f(log |z|). Let F be a nonnegative measurable function of one real variable. Then (a) Z B F ◦φ = an Z 0 −∞ F ◦fe2ntdt (with an being the area of the unit sphere in Cn). and (b) E = 2−n Z 0 −∞

Theorem 4.1. Let φ be a convex function defined in a convex domain Ωin Rn and let ˆφ be its Schwarz symmetrization. Then ˆφ is also convex. This fact should be well known but we include a proof in order to emphazise the similarity with Theorem 2.2. By definition ˆφ(x) = g(|x|) for some increasing function g and we need to prove that g is convex (notice the change in convention as compared with the complex case where we wrote ˆφ(z) = f(log |z|)). As before σ(t) := |{x ∈Ω; φ(x) < t}| = an(g−1(t))n,

Theorem 4.2. Let φ be a convex function in the ball, continuous on the closed ball and vanishing on the boundary. Let ˆφ be its Schwarz symmetrization. Then E(ˆφ) ≤E(φ). This is proved in a way completely parallell to the complex case, so we shall not give the de- tails. We define geodesics and subgeodesics in the space of convex functions as before. Then the real energy is concave along subgeodesics and affine along geodesics as before and the analog of the formula for the first order derivative a

Theorem 4.3. Let Ωbe a bounded convex domain in Rn containing the origin, and let µΩbe the Minkowski functional of Ω. Let u be a convex function in Ωof the form u(x) = f(µΩ(x)), and let ˆu be its Schwarz symmetrization. Then M(Ω)−1EΩ(u) = M(BΩ)−1EB(ˆu) (where BΩis the ball of the same volume as Ω). Notice that we could as well have divided by just |Ω◦| instead of the Mahler volume, since the volumes of Ωand BΩare automatically equal, but the Mahler volume seems to simplify a little below. From t

Theorem 4.4. Let Ωbe a convex domain containing the origin. Assume that for any convex function in Ω, v that vanishes on the boundary the symmetrization inequality M(Ω)−1EΩ(v) ≥M(B)−1EB(ˆv) holds. Then v is an ellipsoid.

Lemma 4.5. Let Ωbe a smoothly bounded convex domain containing the origin, with Minkowski functional µΩ. Let u be a smooth convex function in Ωof the form u(x) = f(µΩ(x)), vanishing on the boundary so that f(1) = 0. Then σ(s) := Z µΩ<s MA(u) = f ′(s)n|Ω◦|.

Lemma 4.6. Under the same hypotheses as in the previous lemma, E(u) = Z 1 f ′(s)n+1ds|Ω◦|.

Lemma 4.7. Let Ωand u be as in the previous lemmas, and let B be a ball centered at the origin of the same volume as Ω. Then SB(u) = f(µB).