The sharp constants in the real anisotropic Littlewood's $\boldsymbol{4 / 3}$ inequality and applications

TL;DR

The paper resolves the sharp real anisotropic Littlewood's 4/3 inequality constants by proving $\|A\|_{a,b} \le 2^{\max\{0,\frac{1}{a}+\frac{1}{b}-1\}} \|A\|$ for all admissible $(a,b)$, yielding $\mathsf{L}_{a,b}^{\mathbb{R}}=2^{\max\{0,\frac{1}{a}+\frac{1}{b}-1\}}$. It also advances the complex case with upper bounds $\mathsf{L}_{a,b}^{\mathbb{C}} \le (4/\pi)^{\max\{0,\frac{1}{a}+\frac{1}{b}-1\}}$ and a partial region-based picture, aided by a new Khinchin-type inequality for Steinhaus variables. A key technical contribution is the equality $\mathsf{S}_r = \mathsf{L}_{1,r}^{\mathbb{C}} = \mathsf{L}_{r,1}^{\mathbb{C}}$, linking complex Littlewood constants to Steinhaus-type Khintchine constants. The authors also derive variants of Khinchin's inequality and use them to bound $(q,s)$-cotype constants of $\ell_1(\mathbb{K})$, illuminating the geometric consequences of anisotropic Littlewood-type inequalities.

Abstract

The real anisotropic Littlewood's $4 / 3$ inequality is an extension of a famous result obtained in 1930 by J. E. Littlewood. It asserts that, for $a , b \in ( 0 , \infty )$, the following conditions are equivalent: $\bullet$ There is an optimal constant $\mathsf{L}_{ a , b }^{ \mathbb{R} } \in [ 1 , \infty )$ such that \[ \Biggl ( \, \sum_{ k = 1 }^{ \infty } \biggl ( \, \sum_{ j = 1 }^{ \infty } \bigl \lvert A \bigl ( \boldsymbol{e}^{ (k) } , \boldsymbol{e}^{ (j) } \bigr ) \bigr \rvert^a \biggr )^{ \frac{b}{a} } \Biggr )^{ \frac{1}{b} } \leq \mathsf{L}_{ a , b }^{ \mathbb{R} } \cdot \lVert A \rVert \] for every continuous bilinear form $A \colon c_0 \times c_0 \to \mathbb{R}$. $\bullet$ The values $a , b$ satisfy $a , b \geq 1$ and $\frac{1}{a} + \frac{1}{b} \leq \frac{3}{2}$. Several authors have obtained the values of $\mathsf{L}_{ a , b }^{ \mathbb{R} }$ for diverse pairs $( a , b )$. In this paper we provide the complete list of such optimal values, as well as new estimates for $\mathsf{L}_{ a , b }^{ \mathbb{C} }$ (the analog for continuous $\mathbb{C}$-bilinear forms), which are exact in several cases. As an application we prove, in terms of the values $\mathsf{L}_{ 1 , r }^{ \mathbb{C} }$, a variant of Khinchin's inequality for Steinhaus variables, and we provide estimates for the optimal $( q , s )$-cotype constants of the spaces $\ell_1 ( \mathbb{K} )$ (with $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$) in terms of the values $\mathsf{L}_{ 1 , q }^{ \mathbb{R} }$.

Figures (3)

• Figure 1: Region N (including the dotted open segments but not other points of its boundary) corresponds to non-admissible exponents in the real and complex anisotropic Littlewood's $4 / 3$ inequality (Proposition \ref{['Lab-weak']} and Remark \ref{['0<s<1;(s,infty)and(infty,s)nonadmissible']}). Region I (closed) corresponds to Remark \ref{['Lab^K=1;ab>=2']}; in this region the optimal constant, both in the real and complex case, has always value $1$. Region II (closed) corresponds to Equation \ref{['Lab^R-incomplete']}, so that the value of the optimal constant in the real case is $2^{ \frac{1}{a} + \frac{1}{b} - 1 }$. Finally, Region III corresponds to Problem \ref{['Problem1']}, whereas the union of Regions II and III minus the two points $( 1 , 2 )$ and $( 2 , 1 )$ corresponds to Problem \ref{['Problem2']}; see Equation \ref{['L_12^C;L_21^C']}.
• Figure 2: Illustration of Theorem \ref{['main-Lab^R']} and its proof (all the regions are closed): Regions RII--RIV correspond to the value $2^{ \frac{1}{a} + \frac{1}{b} - 1 }$ for the optimal constant, whereas Region RI corresponds to points for which the optimal constant is $1$. This settles completely the real case.
• Figure 3: Region CI (closed) consist of points for which the optimal constant in the complex case is $1$ (Theorem \ref{['main-Lab^C']}). Region CII (including the dashed curve minus the points $( 1 , \infty )$ and $( \infty , 1 )$) corresponds to points for which the optimal constant in the complex case is unknown, except for the cases $( a , b ) = ( 1 , 2 )$ and $( a , b ) = ( 2 , 1 )$, for which we have $\mathsf{L} _{ a , b }^{ \varmathbb{C} } = \frac{2}{ {\hbox{$\sqrt[\leftroot{0}\uproot{1}\ ]{ \pi }$} } }$. (Theorem \ref{['main-Lab^C']} provides an upper bound for such unknown values.)

Theorems & Definitions (32)

• proposition 1
• remark 1
• remark 2
• remark 3
• theorem 1
• proposition 2
• lemma 1
• proof
• remark 4
• proposition 3