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March  2011, 10(2): 745-760. doi: 10.3934/cpaa.2011.10.745

The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform

Semyon Yakubovich 1,

1. 

Department of Mathematics, Faculty of Sciences, University of Porto, Campo Alegre st., 687, Porto, 4169-007, Portugal

Received  June 2010 Revised  October 2010 Published  December 2010

We introduce a notion of the heat kernel related to the familiar Kontorovich-Lebedev transform. We study differential and semigroup properties of this kernel and construct fundamental solutions of a generalized diffusion equation. An integral transformation with the heat kernel is considered. By using the Plancherel $L_2$-theory for the Kontorovich-Lebedev transform and norm estimates for its convolution we establish analogs of the classical Heisenberg inequality and uncertainty principle for this transformation. The proof is also based on the norm inequalities for the Mellin transform of the heat kernel.
Keywords: Heisenberg inequality, modi ed Bessel function., Kontorovich-Lebedev transform, the Mellin transform, uncertainty principle, heat kernel, convolution, Plancherel theory.
Mathematics Subject Classification: Primary: 44A15; Secondary: 44A05, 44A3.
Citation: Semyon Yakubovich. The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform. Communications on Pure & Applied Analysis, 2011, 10 (2) : 745-760. doi: 10.3934/cpaa.2011.10.745
References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions," Dover, New York, 1972.  Google Scholar

[2]

R. Ma, Heisenberg inequalities for Jacobi transforms, J. Math. Anal. Appl., 332 (2007), 155-163. doi: doi:10.1016/j.jmaa.2006.09.044.  Google Scholar

[3]

E. H. Lieb and M. Loss, "Analysis," Graduate Studies in Math., Vol. 14, American Math. Soc., Providence, Rhode Island, 2001.  Google Scholar

[4]

C. Monthus and A. Comtet, On the flux distribution in a one dimensional disordered system, J. Phys. I France, 4 (1994), 635-653. Google Scholar

[5]

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series, Vol. I: Elementary Functions," Gordon and Breach, New York and London, 1986.  Google Scholar

[6]

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series, Vol. II: Special Functions," Gordon and Breach, New York and London, 1986.  Google Scholar

[7]

I. N. Sneddon, "The Use of Integral Transforms," McGraw-Hill, New York, 1972. Google Scholar

[8]

S. B. Yakubovich and Yu. F. Luchko, "The Hypergeometric Approach to Integral Transforms and Convolutions," (Kluwers Ser. Math. and Appl.: Vol. 287), Dordrecht, Boston, London, 1994.  Google Scholar

[9]

S. B. Yakubovich, "Index Transforms," World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1996.  Google Scholar

[10]

S. B. Yakubovich, On the Kontorovich-Lebedev transformation, J. of Integral Equations and Appl., 15 (2003), 95-112. doi: doi:10.1216/jiea/1181074947.  Google Scholar

[11]

S. B. Yakubovich, Integral transforms of the Kontorovich-Lebedev convolution type, Collect. Math., 54 (2003), 99-110.  Google Scholar

[12]

S. B. Yakubovich, Boundedness and inversion properties of certain convolution transforms, J. Korean Math. Soc., 40 (2003), 999-1014. doi: doi:10.4134/JKMS.2003.40.6.999.  Google Scholar

[13]

S. B. Yakubovich, On the least values of $L_p$-norms for the Kontorovich-Lebedev transform and its convolution, J. of Approximation Theory, 131 (2004), 231-242. doi: doi:10.1016/j.jat.2004.10.007.  Google Scholar

[14]

S. B. Yakubovich, The Kontorovich-Lebedev transformation on Sobolev type spaces, Sarajevo J. of Mathematics, 1 (2005), 211-234.  Google Scholar

[15]

S. B. Yakubovich, On a testing -function space for distributions associated with the Kontorovich-Lebedev transform, Collect. Math., 57 (2006), 279-293.  Google Scholar

[16]

S. B. Yakubovich, Uncertainty principles for the Kontorovich-Lebedev transform, Math. Modelling and Analysis, 13 (2008), 289-302. doi: doi:10.3846/1392-6292.2008.13.289-302.  Google Scholar

[17]

S. B. Yakubovich, A class of polynomials and discrete transformations associated with the Kontorovich-Lebedev operators, Integral Transforms and Special Functions, 20 (2009), 551-567. doi: doi:10.1080/10652460802648473.  Google Scholar

[18]

S. B. Yakubovich and R. Daher, An analog of Morgan's theorem for the Kontorovich-Lebedev transform, Adv. Pure Apll. Math., 1:2 (2010), 159-162. doi: doi:10.1515/APAM.2010.010.  Google Scholar

[19]

A. H. Zemanian, The Kontorovich-Lebedev transformation on distributions of compact support and its inversion, Math. Proc. Cambridge Philos. Soc., 77 (1975), 139-143. doi: doi:10.1017/S0305004100049471.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions," Dover, New York, 1972.  Google Scholar

[2]

R. Ma, Heisenberg inequalities for Jacobi transforms, J. Math. Anal. Appl., 332 (2007), 155-163. doi: doi:10.1016/j.jmaa.2006.09.044.  Google Scholar

[3]

E. H. Lieb and M. Loss, "Analysis," Graduate Studies in Math., Vol. 14, American Math. Soc., Providence, Rhode Island, 2001.  Google Scholar

[4]

C. Monthus and A. Comtet, On the flux distribution in a one dimensional disordered system, J. Phys. I France, 4 (1994), 635-653. Google Scholar

[5]

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series, Vol. I: Elementary Functions," Gordon and Breach, New York and London, 1986.  Google Scholar

[6]

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series, Vol. II: Special Functions," Gordon and Breach, New York and London, 1986.  Google Scholar

[7]

I. N. Sneddon, "The Use of Integral Transforms," McGraw-Hill, New York, 1972. Google Scholar

[8]

S. B. Yakubovich and Yu. F. Luchko, "The Hypergeometric Approach to Integral Transforms and Convolutions," (Kluwers Ser. Math. and Appl.: Vol. 287), Dordrecht, Boston, London, 1994.  Google Scholar

[9]

S. B. Yakubovich, "Index Transforms," World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1996.  Google Scholar

[10]

S. B. Yakubovich, On the Kontorovich-Lebedev transformation, J. of Integral Equations and Appl., 15 (2003), 95-112. doi: doi:10.1216/jiea/1181074947.  Google Scholar

[11]

S. B. Yakubovich, Integral transforms of the Kontorovich-Lebedev convolution type, Collect. Math., 54 (2003), 99-110.  Google Scholar

[12]

S. B. Yakubovich, Boundedness and inversion properties of certain convolution transforms, J. Korean Math. Soc., 40 (2003), 999-1014. doi: doi:10.4134/JKMS.2003.40.6.999.  Google Scholar

[13]

S. B. Yakubovich, On the least values of $L_p$-norms for the Kontorovich-Lebedev transform and its convolution, J. of Approximation Theory, 131 (2004), 231-242. doi: doi:10.1016/j.jat.2004.10.007.  Google Scholar

[14]

S. B. Yakubovich, The Kontorovich-Lebedev transformation on Sobolev type spaces, Sarajevo J. of Mathematics, 1 (2005), 211-234.  Google Scholar

[15]

S. B. Yakubovich, On a testing -function space for distributions associated with the Kontorovich-Lebedev transform, Collect. Math., 57 (2006), 279-293.  Google Scholar

[16]

S. B. Yakubovich, Uncertainty principles for the Kontorovich-Lebedev transform, Math. Modelling and Analysis, 13 (2008), 289-302. doi: doi:10.3846/1392-6292.2008.13.289-302.  Google Scholar

[17]

S. B. Yakubovich, A class of polynomials and discrete transformations associated with the Kontorovich-Lebedev operators, Integral Transforms and Special Functions, 20 (2009), 551-567. doi: doi:10.1080/10652460802648473.  Google Scholar

[18]

S. B. Yakubovich and R. Daher, An analog of Morgan's theorem for the Kontorovich-Lebedev transform, Adv. Pure Apll. Math., 1:2 (2010), 159-162. doi: doi:10.1515/APAM.2010.010.  Google Scholar

[19]

A. H. Zemanian, The Kontorovich-Lebedev transformation on distributions of compact support and its inversion, Math. Proc. Cambridge Philos. Soc., 77 (1975), 139-143. doi: doi:10.1017/S0305004100049471.  Google Scholar

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