ИСТИНА

Classical Hilbert inequalities assert boundedness of the bilinear forms with Hankel-type Hilbert's matrix A(m,n)=1/(m+n) and Toeplitz-like Cauchy's matrix A(m,n)=1/(m-n) on the space of square-integrable sequences. We introduce a matrix that interpolates between these two cases, A(m,n)=1/(ma+nb), where a and b are distinct complex numbers with modulus 1. This family of bilinear forms is uniformly bounded on l^2. The discrete inequality has a natural integral analog, which is intimately related with a Plancherel-like formula for the Laplace transform along rays in the complex plane. Estimates for the Laplace and Fourier transform along a sufficiently well-behaved curve in the complex plane follow. Using interpolation, the results are extended to Hausdorff-Young type inequalities for Lp norms. This is a joint work with A.Merzon (Mexico).