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September  2012, 11(5): 1629-1642. doi: 10.3934/cpaa.2012.11.1629

Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions

Pablo L. De Nápoli 1, , Irene Drelichman 2, and  Ricardo G. Durán 2,

1. 

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
2. 

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Airess, Argentina, Argentina

Received  September 2010 Revised  November 2011 Published  March 2012

We show that Caffarelli-Kohn-Nirenberg first order interpolation inequalities as well as weighted trace inequalities in $\mathbb{R}^n \times \mathbb{R}_+$ admit a better range of power weights if we restrict ourselves to the space of radially symmetric functions.
Keywords: power weights., Radial functions, Caffarelli-Kohn-Nirenberg inequalities, trace inequalities.
Mathematics Subject Classification: Primary: 26D10, 46E35; Secondary: 47G1.
Citation: Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629
References:
[1]

J. F. Bonder and J. Dolbeault, work in progress, work in progress., ().   Google Scholar

[2]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.  Google Scholar

[3]

F. Catrina and Z-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[4]

P. L. De Nápoli, I. Drelichman and R. G. Durán, On weighted inequalities for fractional integrals of radial functions,, To appear in Illinois J. Math. \arXiv{0910.5508}., ().   Google Scholar

[5]

L. Grafakos, "Classical and Modern Fourier Analysis," Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[6]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[7]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[8]

E. M. Stein and G. Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  Google Scholar

show all references

References:
[1]

J. F. Bonder and J. Dolbeault, work in progress, work in progress., ().   Google Scholar

[2]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.  Google Scholar

[3]

F. Catrina and Z-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[4]

P. L. De Nápoli, I. Drelichman and R. G. Durán, On weighted inequalities for fractional integrals of radial functions,, To appear in Illinois J. Math. \arXiv{0910.5508}., ().   Google Scholar

[5]

L. Grafakos, "Classical and Modern Fourier Analysis," Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[6]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[7]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[8]

E. M. Stein and G. Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  Google Scholar

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