For α > 0, the Bargmann projection P α is the orthogonal projection from L 2 (γ α ) onto the holomorphic subspace L 2 hol (γ α ), where γ α is the standard Gaussian probability measure on C n with variance (2α) −n . The space L 2 hol (γ α ) is classically known as the Segal-Bargmann space. We show that P α extends to a bounded operator on L p (γ αp/2 ), and calculate the exact norm of this scaled L p Bargmann projection.We use this to show that the dual space of the L p -Segal-Bargmann space L p hol (γ αp/2 ) is an L p SegalBargmann space, but with the Gaussian measure scaled differently: (L p hol (γ αp/2 )) * ∼ = L p hol (γ αp /2 ) (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms.

Narasimhan and Ramadas showed in [16] that the Gribov ambiguity was maximal for the product SU (2) bundle over S 3 . Specifically they showed that the holonomy group of the Coulomb connection is dense in the gauge group. Instead of base manifold S 3 , we consider here a base manifold with a boundary. In this with-boundary case we must include boundary conditions on the connection forms. We will use the so-called conductor boundary conditions on connections. With these boundary conditions, we will first show that the space of connections is a C ∞ Hilbert principal bundle with respect to the associated conductor gauge group. We will consider the holonomy of the Coulomb connection for this bundle. If the base manifold is an open subset of R 3 and we use the product principal bundle, we will show that the holonomy group is again a dense subset of the gauge group.We will be concerned with boundary conditions on p-forms that have been dubbed conductor boundary conditions by Gross in [8]. We say a form ω satisfies conductor boundary conditions if i * (ω) = 0 where i : ∂M → M is the inclusion map. In other words ω(X 1 ∧ . . . ∧ X p ) = 0 if X 1 , . . . , X p are all tangent to the boundary. This is half of the "relative" boundary conditions given by Ray and Singer in [19], and the boundary conditions of the "Dirichlet problem" of Marini in [13].We can extend the notion of conductor boundary conditions to connections. Given an Ehresmann connection ω A on P , we consider the induced Koszul connection ∇ A on the vector bundle E. Given a fixed connection A 0 on P , we say that a connection ∇ A satisfies conductor boundary condtions with respect to ∇ A0 if the 1-form ∇ A − ∇ A0 satisfies conductor boundary conditions. Such a ∇ A equals ∇ A0 on the boundary in tangential directions, giving us a Dirichletlike boundary condition (hence the terminology found in [13]). If we restrict our view to connections satisfying conductor boundary conditions with respect to a fixed A 0 , we must change the gauge group so that it preserves the boundary conditions. A gauge transformation g ∈ G is a section of certain bundle over M with fibers diffeomorphic to K. The conductor gauge group G con consists of those gauge transformations g such that g| ∂M ≡ e, where e is the identity of K. Note that the definition of G con does not depend on the fixed connection ∇ A0 .

We study dilated holomorphic L p space of Gaussian measures over C n , denoted H n p,α with variance scaling parameter α > 0. The duality relations (H n p,α ) * ∼ = H p ′ ,α hold with 1 p + 1 p ′ = 1, but not isometrically. We identify the sharp lower constant comparing the norms on H p ′ ,α and (H n p,α ) * , and provide upper and lower bounds on the sharp upper constant. We prove several suggestive partial results on the sharpness of the upper constant. One of these partial results leads to a sharp bound on each Taylor coefficient of a function in the Fock space for n = 1.

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We theoretically and experimentally analyze recent changes in the posting system used by Japan's Nippon Professional Baseball (NPB) organization and the U.S. Major League Baseball (MLB) organization to transfer the rights of NPB players from NPB teams to MLB teams. Under the old system-a sequential, first-price sealed-bid auction among MLB teams for player negotiation exclusivity rights-the NPB team enjoyed considerable surplus from lucrative posting fees. We predict that the revised system-an English auction with an entry fee-will transfer most of the NPB's middleman benefits (posting fees) to the players through higher salaries. Additionally, we analyze a third, proposed but not adopted, mechanism-a weighted-average sealed-bid auction. Our experimental results confirm our theoretical predictions. Furthermore, under the new system, efficiency is greatest, MLB teams spend less on Japanese players, and the players' salaries increase significantly.