Littlewood's 4/3 inequality

In mathematical analysis, Littlewood's 4/3 inequality, named after John Edensor Littlewood,^[1] is an inequality that holds for every complex-valued bilinear form defined on ${\displaystyle c_{0}}$, the Banach space of scalar sequences that converge to zero.

Precisely, let ${\displaystyle B:c_{0}\times c_{0}\to \mathbb {C} }$ or ${\displaystyle \mathbb {R} }$ be a bilinear form. Then the following holds:

${\displaystyle \left(\sum _{i,j=1}^{\infty }|B(e_{i},e_{j})|^{4/3}\right)^{3/4}\leq {\sqrt {2}}\|B\|,}$

where

${\displaystyle \|B\|=\sup\{|B(x_{1},x_{2})|:\|x_{i}\|_{\infty }\leq 1\}.}$

The exponent 4/3 is optimal, i.e., cannot be improved by a smaller exponent.^[2] It is also known that for real scalars the aforementioned constant is sharp.^[3]

Generalizations

Bohnenblust–Hille inequality

Bohnenblust–Hille inequality^[4] is a multilinear extension of Littlewood's inequality that states that for all ${\displaystyle m}$-linear mapping ${\displaystyle M:c_{0}\times \cdots \times c_{0}\to \mathbb {C} }$ the following holds:

${\displaystyle \left(\sum _{i_{1},\ldots ,i_{m}=1}^{\infty }|M(e_{i_{1}},\ldots ,e_{i_{m}})|^{2m/(m+1)}\right)^{(m+1)/(2m)}\leq 2^{(m-1)/2}\|M\|,}$

See also

• Grothendieck inequality

References

1. Littlewood, J. E. (1930). "On bounded bilinear forms in an infinite number of variables". The Quarterly Journal of Mathematics. os-1 (1): 164–174. Bibcode:1930QJMat...1..164L. doi:10.1093/qmath/os-1.1.164.
2. Littlewood, J. E. (1930). "On bounded bilinear forms in an infinite number of variables". The Quarterly Journal of Mathematics (1): 164–174. Bibcode:1930QJMat...1..164L. doi:10.1093/qmath/os-1.1.164.
3. Diniz, D. E.; Munoz, G.; Pellegrino, D.; Seoane, J. (2014). "Lower bounds for the Bohnenblust--Hille inequalities: the case of real scalars". Proceedings of the American Mathematical Society (132): 575–580. arXiv:1111.3253. doi:10.1090/S0002-9939-2013-11791-0. S2CID 119128323.
4. Bohnenblust, H. F.; Hille, Einar (1931). "On the Absolute Convergence of Dirichlet Series". The Annals of Mathematics. 32 (3): 600–622. doi:10.2307/1968255. JSTOR 1968255.