Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

[EN] By the von Neumann inequality for homogeneous polynomials there exists a positive constant C-k,C-q(n) such that for every k-homogeneous polynomial p in n variables and every n-tuple of commuting operators (T-1, ..., T-n with Sigma(n)(i=1) parallel to T-i parallel to(q) <= 1 we have parallel to P (T-1, ..., T-n)parallel to L(H) <= C-k,C-q(n) sup {vertical bar p(z(1), ..., z(n))vertical bar: Sigma(n)(i=1) vertical bar(q) <= 1}. For fixed k and q, we study the asymptotic growth of the smallest constant C-k,C-q(n) as n (the number of variables/operators) tends to infinity. For q = infinity, we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the 1970s). For 2 <= q < infinity we improve some lower bounds given by Mantero and Tonge, and prove the asymptotic behavior up to a logarithmic factor. To achieve this we provide estimates of the norm of homogeneous unimodular Steiner polynomials, i.e. polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems.The first two named authors were supported by CONICET projects PIP 0624 and PICT 2011-1456, and by UBACyT projects 20020130300057BA and 20020130300052BA. The third named author was supported by MICINN project MTM2014-57838-C2-2-P.Galicer, D.; Muro, S.; Sevilla Peris, P. (2018). Asymptotic estimates on the von Neumann inequality for homogeneous polynomials. Journal für die reine und angewandte Mathematik (Crelles Journal). 743:213-227. https://doi.org/10.1515/crelle-2015-0097S213227743Defant, A., Garcia, D., & Maestre, M. (2004). MAXIMUM MODULI OF UNIMODULAR POLYNOMIALS. Journal of the Korean Mathematical Society, 41(1), 209-229. doi:10.4134/jkms.2004.41.1.209Bayart, F. (2010). MAXIMUM MODULUS OF RANDOM POLYNOMIALS. The Quarterly Journal of Mathematics, 63(1), 21-39. doi:10.1093/qmath/haq026Crabb, M. J., & Davie, A. M. (1975). Von Neumann’s Inequality for Hilbert Space Operators. Bulletin of the London Mathematical Society, 7(1), 49-50. doi:10.1112/blms/7.1.49Alon, N., Kim, J.-H., & Spencer, J. (1997). Nearly perfect matchings in regular simple hypergraphs. Israel Journal of Mathematics, 100(1), 171-187. doi:10.1007/bf02773639Defant, A., Frerick, L., Ortega-Cerdà, J., Ounaïes, M., & Seip, K. (2011). The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Annals of Mathematics, 174(1), 485-497. doi:10.4007/annals.2011.174.1.13Maurizi, B., & Queffélec, H. (2009). Some Remarks on the Algebra of Bounded Dirichlet Series. Journal of Fourier Analysis and Applications, 16(5), 676-692. doi:10.1007/s00041-009-9112-yCarando, D., & Dimant, V. (2006). Extension of polynomials and John’s theorem for symmetric tensor products. Proceedings of the American Mathematical Society, 135(6), 1769-1773. doi:10.1090/s0002-9939-06-08666-7Blei, R. C. (1979). Multidimensional extensions of the Grothendieck inequality and applications. Arkiv för Matematik, 17(1-2), 51-68. doi:10.1007/bf02385457Schütt, C. (1984). Entropy numbers of diagonal operators between symmetric Banach spaces. Journal of Approximation Theory, 40(2), 121-128. doi:10.1016/0021-9045(84)90021-2Mantero, A., & Tonge, A. (1980). The Schur multiplication in tensor algebras. Studia Mathematica, 68(1), 1-24. doi:10.4064/sm-68-1-1-24Varopoulos, N. T. (1974). On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory. Journal of Functional Analysis, 16(1), 83-100. doi:10.1016/0022-1236(74)90071-8Rödl, V. (1985). On a Packing and Covering Problem. European Journal of Combinatorics, 6(1), 69-78. doi:10.1016/s0195-6698(85)80023-

A dimension-free Remez-type inequality on the polytorus

Consider $f:\Omega^n_K\to\mathbf{C}$ a function from the $n$-fold product of multiplicative cyclic groups of order $K$. Any such $f$ may be extended via its Fourier expansion to an analytic polynomial on the polytorus $\mathbf{T}^n$, and the set of such polynomials coincides with the set of all analytic polynomials on $\mathbf{T}^n$ of individual degree at most $K-1$. In this setting it is natural to ask how the supremum norms of $f$ over $\mathbf{T}^n$ and over $\Omega_K^n$ compare. We prove the following Remez-type inequality: if $f$ has degree at most $d$ as an analytic polynomial, then $\|f\|_{\mathbf{T}^n}\leq C(d,K)\|f\|_{\Omega_K^n}$ with $C(d,K)$ independent of dimension $n$. As a consequence we also obtain a new proof of the Bohnenblust--Hille inequality for functions on products of cyclic groups. Key to our argument is a special class of Fourier multipliers on $\Omega_K^n$ which are $L^\infty\to L^\infty$ bounded independent of dimension when restricted to low-degree polynomials. This class includes projections onto the $k$-homogeneous parts of low-degree polynomials as well as projections of much finer granularity.Comment: 21 pages. Largely revise