Heat-flow Monotonicity Related to the Hausdorff–young Inequality

It is known that if q is an even integer then the L(R) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres “simultaneously slide” to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if q > 2 is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff–Young inequality. For d ∈ N we let Ht denote the heat kernel on R given by Ht(x) = t e 2/t, and we define the Fourier transform μ̂ of a finite Borel measure μ on R by μ̂(ξ) := ∫ Rd e dμ(x). In what follows, for p ∈ [1,∞], p will denote the dual exponent satisfying 1 p+ 1 p = 1. For μ a positive finite Borel measure on R and 2 ≤ q ≤ p ≤ ∞, let Qp,q : (0,∞) → R be given by Qp,q(t) = t d 2 “ 1 q − 1 p ” ∥∥ ̂ u(t, ·)1/p ∥∥ q , where u(t, ·) = Ht ∗ μ. If q = 2k is an even integer then by Plancherel’s theorem one may write Qp,q in terms of a k-fold convolution (1) Qp,q(t) = t d 2 “