On the least values of [formula omitted]-norms for the Kontorovich–Lebedev transform and its convolution

Abstract

We establish analogs of the Hausdorff–Young and Riesz–Kolmogorov inequalities and the norm estimates for the Kontorovich–Lebedev transformation and the corresponding convolution. These classical inequalities are related to the norms of the Fourier convolution and the Hilbert transform in L p spaces, 1 ⩽ p ⩽ ∞ . Boundedness properties of the Kontorovich–Lebedev transform and its convolution operator are investigated. In certain cases the least values of the norm constants are evaluated. Finally, it is conjectured that the norm of the Kontorovich–Lebedev operator K i τ : L p ( R + ; x dx ) → L p ( R + ; x sinh π x dx ) , 2 ⩽ p ⩽ ∞ K i τ [ f ] = ∫ 0 ∞ K i τ ( x ) f ( x ) dx , τ ∈ R + is equal to π 2 1 - 1 p . It confirms, for instance, by the known Plancherel-type theorem for this transform when p = 2 .