ИСТИНА

The motivation of the main result came from a concrete computational question: to determine the best constant in Hardy's inequality for the norm of the Laplace transform operator (LT) in Lebesgue spaces. For p>=1, let N(p) be the norm of the operator LT acting from Lp(R+) to Lp'(R+). The classical Hardy inequality N(p)<= (2\pi/p')^{1/p'} is proved by reduction to Young's convolution norm inequality. The exact value of N(p) in closed form is unknown except in the cases p=1,2,infinity. (For the Fourier transform, which can be thought of as a Laplace transform evaluated along the imaginary axis, the exact norm is known due to Babenko and Beckner.) The constant N(2) follows by the explicit spectral analysis of the operator |LT|^2 in L2. Its spectrum has no atoms; in particular, there is no maximizer. The corresponding convolution operator from L2 to L2 in Hardy's reduction also does not have a maximizer. The situation is opposite for convolution operators with kernels in Lq acting from Lp to Lr, where 1/p+1/q=1+1/r and neither of p, q, r equals 1 or infinity. (The boundedness of any such operator follows from Young's inequality.) In our paper (Sbornik Math.,2019) we proved that the maximizer of convolution in this situation always exists. As a consequence, constructing approximations converging to a maximizer becomes a feasible approach for a numerical calculation of the convolution norm. The above defined norms N(p) have been thus computed. A weaker version of the maximizer existence theorem (Pearson,1999) was insufficient for this application. Some related questions, resolved and open, are to be discussed. We will mention major advances bordering the present topic, reported by M. Christ and his school in the 2010s.