Convolutions related to the Fourier and Kontorovich–Lebedev transforms revisited

Abstract

We continue to investigate boundedness properties in a two-parametric family of Lebesgue spaces for convolutions related to the Fourier and Kontorovich–Lebedev transforms. Norm estimations in the weighted L _p -spaces are obtained and applications to the corresponding class of convolution integral equations are demonstrated. Necessary and sufficient conditions are found for the solvability of these equations in the weighted L ₂-spaces.

Keywords:

• Fourier transform
• Kontorovich–Lebedev transform
• modified Bessel function
• convolution integral equations
• Plancherel theorem

2000 Mathematics Subject Classification :

• 44A15
• 44A05
• 44A35
• 33C10
• 45A05

Acknowledgements

The present investigation was partially supported by the ‘Centro de Matemática’ of the University of Porto.