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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 and 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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

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