Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L²(ℝ₊;dx/x)

References

[1] P. K. Banerji, D. Loonker and S. L. Kalla, Kontorovich–Lebedev transform for Boehmians, Integral Transforms Spec. Funct. 20 (2009), no. 12, 905–913. 10.1080/10652460902987060Search in Google Scholar

[2] A. Erde’lyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. Vol. 2, McGraw-Hill, New York, 1953. Search in Google Scholar

[3] H. J. Glaeske and A. Heß, A convolution connected with the Kontorovich–Lebedev transform, Math. Z. 193 (1986), no. 1, 67–78. 10.1007/BF01163354Search in Google Scholar

[4] B. J. González and E. R. Negrín, Operational calculi for Kontorovich–Lebedev and Mehler–Fock transforms on distributions with compact support, Rev. Colombiana Mat. 32 (1998), no. 1, 81–92. Search in Google Scholar

[5] N. T. Hong, P. V. Hoang and V. K. Tuan, The convolution for the Kontorovich–Lebedev transform revisited, J. Math. Anal. Appl. 440 (2016), no. 1, 369–378. 10.1016/j.jmaa.2016.03.052Search in Google Scholar

[6] L. Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), no. 3, 501–517. 10.1002/cpa.3160180307Search in Google Scholar

[7] J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), no. 1–2, 269–305. 10.1002/cpa.3160180121Search in Google Scholar

[8] M. I. Kontorovich and N. N. Lebedev, On the one method of solution for some problems in diffraction theory and related problems (in Russian), J. Exp. Theor. Phys. 8 (1938), no. 10–11, 1192–1206. Search in Google Scholar

[9] M. I. Kontorovich and N. N. Lebedev, On the application of inversion formulae to the solution of some electrodynamics problems (in Russian), J. Exp. Theor. Phys. 9 (1939), no. 6, 729–742. Search in Google Scholar

[10] N. N. Lebedev, Analog of the Parseval theorem for the one integral transform (in Russian), Dokl. Akad. Nauk SSSR 68 (1949), no. 3, 653–656. Search in Google Scholar

[11] J. S. Lowndes, An application of the Kontorovich–Lebedev transform, Proc. Edinb. Math. Soc. (2) 11 (1959), no. 3, 135–137. 10.1017/S0013091500021593Search in Google Scholar

[12] J. S. Lowndes, Parseval relations for Kontorovich–Lebedev transform, Proc. Edinburgh Math. Soc. 13 (1962), no. 1, 5–11. 10.1017/S0013091500014437Search in Google Scholar

[13] R. S. Pathak, Pseudo-differential operator associated with the Kontorovich–Lebedev transform, Invest. Math. Sci. 5 (2015), no. 1, 29–46. Search in Google Scholar

[14] R. S. Pathak and J. N. Pandey, The Kontorovich–Lebedev transformation of distributions, Math. Z. 165 (1979), no. 1, 29–51. 10.1007/BF01175128Search in Google Scholar

[15] R. S. Pathak and S. K. Upadhyay, Pseudo-differential operators involving Hankel transforms, J. Math. Anal. Appl. 213 (1997), no. 1, 133–147. 10.1006/jmaa.1997.5495Search in Google Scholar

[16] A. Prasad and V. K. Singh, Pseudo-differential operators associated to a pair of Hankel–Clifford transformations on certain Beurling type function spaces, Asian-Eur. J. Math. 6 (2013), no. 3, Article ID 1350039. 10.1142/S1793557113500393Search in Google Scholar

[17] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series. Vol. I: Elementary Functions, Gordon and Breach Science, Amsterdam, 1986. Search in Google Scholar

[18] A. P. Prudnikov and O. I. Marichev, Integrals and Series: Special Functions. Vol. 2, CRC Press, Boca Raton, 1998. Search in Google Scholar

[19] Y. M. Rappoport, Integral equations and Parseval equalities for the modified Kontorovich–Lebedev transforms, Differ. Uravn. 17 (1981), no. 9, 1697–1699. Search in Google Scholar

[20] L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, Singapore, 1993. 10.1142/1550Search in Google Scholar

[21] N. B. Salem and A. Dachraoui, Pseudo-differential operator associated with the Jacobi differential operator, J. Math. Anal. Appl. 220 (1998), no. 1, 365–381. 10.1006/jmaa.1997.5891Search in Google Scholar

[22] M. A. Salem, A. H. Kamel and H. Bagci, On the use of Kontorovich–Lebedev transform in electromagnetic diffraction by an impedance cone, 2012 International Conference onMathematical Methods in Electromagnetic Theory (MMET), IEEE Press, Piscataway (2012), 10.1109/MMET.2012.6331261. 10.1109/MMET.2012.6331261Search in Google Scholar

[23] I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972. Search in Google Scholar

[24] M. W. Wong, An Introduction to Pseudo-Differential Operators, 3rd ed., World Scientific, Singapore, 2014. 10.1142/9074Search in Google Scholar

[25] S. B. Yakubovich, Index Transforms, World Scientific, Singapore, 1996. 10.1142/2707Search in Google Scholar

[26] S. B. Yakubovich, On the least values of Lp-norms for the Kontorovich–Lebedev transform and its convolution, J. Approx. Theory 131 (2004), no. 2, 231–242. 10.1016/j.jat.2004.10.007Search in Google Scholar

[27] S. B. Yakubovich, The Kontorovich–Lebedev transformation on Sobolev type spaces, Sarajevo J. Math. 1 (2005), no. 14, 211–234. Search in Google Scholar

[28] S. B. Yakubovich and L. E. Britvina, A convolution related to the inverse Kontorovich–Lebedev transform, Sarajevo J. Math. 3 (2007), no. 16, 215–232. Search in Google Scholar

[29] S. B. Yakubovich and J. D. Graff, On parseval equalities and boundedness properties for Kontorovich–Lebedev type operators, Novi Sad J. Math. 29 (1999), no. 1, 185–205. Search in Google Scholar

[30] S. B. Yakubovich and Y. F. Luckho, The Hypergeometric Approach to Integral Transforms and Convolutions, Math. Appl. 287, Kluwer Academic, Dordrecht, 2012. Search in Google Scholar

[31] S. Zaidman, Pseudo-differential operators, Ann. Mat. Pura Appl. (4) 92 (1972), no. 1, 345–399. 10.1007/BF02417953Search in Google Scholar