Integrals and series of special functions
Integrals and series of special functions
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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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.