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

[摘要] 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 less than or equal to p less than or equal to infinity. 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-itau : L-p(R+; x dx) --> L-p(R+; x sinh pix dx), 2 less than or equal to p less than or equal to infinity K-itau[f] = integral(0)(infinity) K-itau(x)f(x)dx, tauis an element ofR(+) is equal to pi/2(1-1/p) It confirms, for instance, by the known Plancherel-type theorem for this transform when p = 2. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-01

[关键词] Kontorovich-Lebedev transform;convolution;Hausdorff-Young inequality;Riesz-Kolmogorov inequality;Fourier transform;Hilbert transform;Plancherel theory;modified Bessel function