On an Extremal Problem Originating in Questions of Unconditional Convergence
Hermann König · Birkhäuser Basel eBooks · 2001
Let n ∈ ℕ and define K : ℝ n × ℝ n → ℝ by $$ K\left( {x,y} \right): = \sin \left( { is the standard scalar product on ℝ n . Define T K : L ∞(∝ n )→ L 1(ℝ n ) by $${T_K}f(x) = \int\limits_{{\mathbb{R}^n}} {K(x,y)f(y)dy, x \in {\mathbb{Z}^n}} $$ i.e. T K is the exponentially weighted odd part of the Fourier transform on ℝ n . Is it true that the operator norm of T K is attained on functions like sgn x 1, i.e. is (1.1) $${\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})}}$$ for any dimension n?