Hilbert inequality

A theorem of D. Hilbert on double series:

$$ \tag{* } \sum _ {m = 1 } ^ \infty \sum _ {n = 1 } ^ \infty \frac{a _ {n} b _ {m} }{n + m } < \ \frac \pi {\sin ( \pi /p) } \left ( \sum _ {n = 1 } ^ \infty a _ {n} ^ {p} \right ) ^ {1/p} \ \left ( \sum _ {m = 1 } ^ \infty b _ {m} ^ {q } \right ) ^ {1/q } , $$

where

$$ p > 1,\ \ q = \frac{p}{p - 1 } ,\ \ { \frac{1}{p} } + { \frac{1}{q } } = 1,\ \ a _ {n} , b _ {m} \geq 0, $$

and the series on the right-hand side have finite positive sums. The constant $ \pi / {\sin ( \pi / p ) } $ is precise, i.e. it cannot be decreased. The validity of (*) with $ p= 2 $ was demonstrated by Hilbert, without the precise constant, in his course on integral equations. Its proof was published by H. Weyl [1]. The precise constant was found by I. Schur [2], while the inequality (*) for arbitrary $ p > 1 $ was first quoted by G.H. Hardy and M. Riesz in 1925. There exist integral analogues and generalizations of (*), for example

$$ \int\limits _ { 0 } ^ \infty \int\limits _ { 0 } ^ \infty K ^ \lambda ( x, y) f ( x) g ( y) dx dy \leq $$

$$ \leq \ K ^ \lambda \left ( \int\limits _ { 0 } ^ \infty f ^ { p } ( x) dx \right ) ^ {1/p} \left ( \int\limits _ { 0 } ^ \infty g ^ {r} ( y) dy \right ) ^ {1/r} , $$

where $ K( x, y) $ is a non-negative kernel, homogeneous of degree $ - 1 $, $ p > 1 $, $ r > 1 $, $ \lambda = p ^ {-} 1 + r ^ {-} 1 \leq 1 $, $ f, g \geq 0 $, and

$$ K = \int\limits _ { 0 } ^ \infty u ^ {- 1/ \lambda q } K ( 1, u) du; $$

and the previously obtained special case of this inequality [4] with kernel $ K( x, y) = 1/( x + y) $( the so-called double-parametric Hilbert inequality) and constant $ K ^ \lambda = ( \pi / \sin {\lambda q } ) ^ \lambda $. The preciseness of this constant has been proved for $ r/( r- 1) = p $. It is also asymptotically precise as $ p \rightarrow 1 $ for an arbitrary admissible fixed $ r $. The problem of the asymptotic behaviour of the constant in (*) for finite sums ( $ 1 \leq n, m \leq N $) has not been solved (1988); it is only known that if $ p = q = 2 $, the constant is

$$ \pi - \frac{\pi ^ {5} }{2} ( \mathop{\rm ln} N) ^ {2} + O ( \mathop{\rm ln} \mathop{\rm ln} \{ N \ ( \mathop{\rm ln} N) ^ {-} 3 \} ). $$

References

[1]	H. Weyl, "Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems" , Göttingen (1908) (Thesis)
[2]	I. Schur, "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen" J. Reine Angew. Math. , 140 (1911) pp. 1–28
[3]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[4]	F.F. Bonsall, "Inequalities with non-conjugate parameters" Quart. J. Math. Oxford (2) , 2 (1951) pp. 135–150
[5]	V. Levin, "On the two-parameter extension and analogue of Hilbert's inequality" J. London Math. Soc. (1) , 11 (1936) pp. 119–124
[6]	N.G. de Bruijn, H.S. Wilf, "On Hilbert's inequality in dimensions" Bull. Amer. Math. Soc. , 68 (1962) pp. 70–73
[7]	P.L. Walker, "A note on an inequality with non-conjugate parameters" Proc. Edinburgh Math. Soc. , 18 (1973) pp. 293–294