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Littlewood's 4/3 inequality

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

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

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

where

‖ B ‖ = sup { | B ( x 1 , x 2 ) | : ‖ x i ‖ ∞ ≤ 1 } . {\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. It is also known that for real scalars the aforementioned constant is sharp.

Generalizations

Bohnenblust–Hille inequality

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

( ∑ i 1 , … , i m = 1 ∞ | M ( e i 1 , … , e i m ) | 2 m / ( m + 1 ) ) ( m + 1 ) / ( 2 m ) ≤ 2 ( m − 1 ) / 2 ‖ M ‖ , {\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

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.


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