On an Extremal Problem Originating in Questions of Unconditional Convergence

Abstract

Let n ∈ ℕ and define K : ℝⁿ × ℝⁿ → ℝ by

$$ K\left( {x,y} \right): = \sin \left( { < x,y} \right)\exp \left( { - \left( {{{\left\| x \right\|}^2} + {{\left\| y \right\|}^2}} \right)/2} \right);x,y \in R{^n} $$

where ‖·‖ is the euclidean norm and < ·, · > is the standard scalar product on ℝⁿ. Define T _K : L _∞(∝ⁿ)→ L ₁(ℝⁿ) by

$${T_K}f(x) = \int\limits_{{\mathbb{R}^n}} {K(x,y)f(y)dy, x \in {\mathbb{Z}^n}} $$

i.e. T _K is the exponentially weighted odd part of the Fourier transform on ℝⁿ. Is it true that the operator norm of T _K is attained on functions like sgn x ₁, i.e. is

$${\left\| {{T_K}:{L_\infty }({\mathbb{R}^n}) \to {L_\infty }({\mathbb{R}^n})} \right\|_{Op}} = {\left\| {T(\operatorname{sgn} {x_1})} \right\|_{{L_1}({\mathbb{R}^n})}}$$

((1.1))

for any dimension n?