The geometry of L(3l2∞) and optimal constants in the Bohnenblust-Hille inequality for multilinear forms and polynomials
doi:10.17398/2605-5686.33.1.51
Keywords:
Extreme points, exposed points, the optimal constants in the Bohnenblust-Hille inequality for symmetric multilinear forms and polynomialsAbstract
We classify the extreme and exposed 3-linear forms of the unit ball of L(3l∞2). We introduce optimal constants in the Bohnenblust-Hille inequality for symmetric multilinear forms and polynomials and investigate about their relations.
Downloads
References
R.M. Aron, Y.S. Choi, S.G. Kim, M. Maestre, Local properties of polynomials on a Banach space, Illinois J. Math. 45 (1) (2001), 25 – 39.
C. Boyd, R.A. Ryan, Geometric theory of spaces of integral polynomials and symmetric tensor products, J. Funct. Anal. 179 (1) (2001), 18 – 42.
W. Cavalcante, D. Pellegrino, Geometry of the closed unit ball of the space of bilinear forms on l∞ 2, arXiv:1603.01535v2.
Y.S. Choi, H. Ki, S.G. Kim, Extreme polynomials and multilinear forms on l1 , J. Math. Anal. Appl. 228 (2) (1998), 467 – 482.
Y.S. Choi, S.G. Kim, The unit ball of P(2 l2 2), Arch. Math. (Basel) 71 (6) (1998), 472 – 480.
Y.S. Choi, S.G. Kim, Extreme polynomials on c0, Indian J. Pure Appl. Math. 29 (10) (1998), 983 – 989.
Y.S. Choi, S.G. Kim, Smooth points of the unit ball of the space P(2 l1 ), Results Math. 36 (1-2) (1999), 26 – 33.
Y.S. Choi, S.G. Kim, Exposed points of the unit balls of the spaces P(2 lp 2) (p = 1, 2, ∞), Indian J. Pure Appl. Math. 35 (1) (2004), 37 – 41.
V. Dimant, D. Galicer, R. Garcı́a, Geometry of integral polynomials, M -ideals and unique norm preserving extensions, J. Funct. Anal. 262 (5) (2012), 1987 – 2012.
S. Dineen, “Complex Analysis on Infinite Dimensional Spaces”, Springer- Verlag, London, 1999.
S. Dineen, Extreme integral polynomials on a complex Banach space, Math. Scand. 92 (1) (2003), 129 – 140.
D. Diniz, G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane- Sepúlveda, Lower bounds for the constants in the Bohnenblust-Hille in- equality: the case of real scalars, Proc. Amer. Math. Soc. 142 (2) (2014), 575 – 580.
B.C. Grecu, Geometry of 2-homogeneous polynomials on lp spaces, 1 < p < ∞, J. Math. Anal. Appl. 273 (2) (2002), 262 – 282.
B.C. Grecu, G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda, Unconditional constants and polynomial inequalities, J. Approx. Theory 161 (2) (2009), 706 – 722.
S.G. Kim, Exposed 2-homogeneous polynomials on P(2 lp 2) (1 ≤ p ≤ ∞), Math. Proc. R. Ir. Acad. 107A (2) (2007), 123 – 129.
S.G. Kim, The unit ball of Ls (2l∞), 2 Extracta Math. 24 (1) (2009), 17 – 29.
S.G. Kim, The unit ball of P(2 d∗(1, w)2 ), Math. Proc. R. Ir. Acad. 111A (2) (2011), 79 – 94.
S.G. Kim, The unit ball of Ls (2d∗ (1, w)2), Kyungpook Math. J. 53 (2) (2013), 295 – 306.
S.G. Kim, Smooth polynomials of P(2d∗ (1, w)2 ), Math. Proc. R. Ir. Acad. 113A (1) (2013), 45 – 58.
S.G. Kim, Extreme bilinear forms of L(2 d∗(1, w)2 ), Kyungpook Math. J. 53 (4) (2013), 625 – 638.
S.G. Kim, Exposed symmetric bilinear forms of Ls(2 d∗(1, w)2 ), Kyungpook Math. J. 54 (3) (2014), 341 – 347.
S.G. Kim, Exposed bilinear forms of L(2 d∗(1, w)2 ), Kyungpook Math. J. 55 (1) (2015), 119 – 126.
S.G. Kim, Exposed 2-homogeneous polynomials on the 2-dimensional real predual of Lorentz sequence space, Mediterr. J. Math. 13 (5) (2016), 2827 – 2839.
S.G. Kim, The geometry of L(2l∞ 2 ), to appear in Kyungpook Math. J. 58 (2018).
S.G. Kim, The unit ball of Ls(2 l∞ 3 ), Comment. Math. Prace Mat. 57 (1) (2017), 1 – 7.
S.G. Kim, The geometry of Ls(3 l∞ 2 ), Commun. Korean Math. Soc. 32 (4) (2017), 991 – 997.
S.G. Kim, S.H. Lee, Exposed 2-homogeneous polynomials on Hilbert spaces, Proc. Amer. Math. Soc. 131 (2) (2003), 449 – 453.
J. Lee, K.S. Rim, Properties of symmetric matrices, J. Math. Anal. Appl. 305 (1) (2005), 219 – 226.
G.A. Muñoz-Fernández, S. Révész, J.B. Seoane-Sepúlveda, Geometry of homogeneous polynomials on non symmetric convex bodies, Math. Scand. 105 (1) (2009), 147 – 160.
G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda, Geometry of Banach spaces of trinomials, J. Math. Anal. Appl. 340 (2) (2008), 1069 – 1087.
R.A. Ryan, B. Turett, Geometry of spaces of polynomials, J. Math. Anal. Appl. 221 (2) (1998), 698 – 711.
W.M. Ruess, C.P. Stegall, Extreme points in duals of operator spaces, Math. Ann. 261 (4) (1982), 535 – 546.
