On the least values of Lp-norms for the Kontorovich–Lebedev transform and its convolution

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 Lp 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 Ki τ : Lp (R+;xdx) → Lp(R+;x sinh πx dx), 2 ≤ p ≤ ∞ Kiτ[f] = ∫0∞ Kiτ(x)f(x)dx, τ ∈ R+ is equal to π/21-1/p. It confirms, for instance, by the known Plancherel-type theorem for this transform when p = 2.