American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Regularizing rate estimates for mild solutions of the incompressible Magneto-hydrodynamic system
  • CPAA Home
  • This Issue
  • Next Article
    Positive solutions of a fourth-order boundary value problem involving derivatives of all orders
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. 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. 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, (2004). Google Scholar

[6]

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

[7]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970). Google Scholar

[8]

E. M. Stein and G. Weiss, Fractional integrals on $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. 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. 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. 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, (2004). Google Scholar

[6]

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

[7]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970). Google Scholar

[8]

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

[1]

Mayte Pérez-Llanos. Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 1975-2005. doi: 10.3934/cpaa.2016024
[2]

B. Abdellaoui, I. Peral. On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2003, 2 (4) : 539-566. doi: 10.3934/cpaa.2003.2.539
[3]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002
[4]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697
[5]

Stathis Filippas, Luisa Moschini, Achilles Tertikas. Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (2) : 373-382. doi: 10.3934/cpaa.2015.14.373
[6]

Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39
[7]

Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116
[8]

Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111
[9]

Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599
[10]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683
[11]

Minghua Li, Chunrong Chen, Shengjie Li. Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019001
[12]

Anuradha Sharma, Saroj Rani. Trace description and Hamming weights of irreducible constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 123-141. doi: 10.3934/amc.2018008
[13]

Vy Khoi Le. Existence and enclosure of solutions to noncoercive systems of inequalities with multivalued mappings and non-power growths. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 255-276. doi: 10.3934/dcds.2013.33.255
[14]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control & Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365
[15]

Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695
[16]

Jean Dolbeault, Maria J. Esteban, Gaspard Jankowiak. Onofri inequalities and rigidity results. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3059-3078. doi: 10.3934/dcds.2017131
[17]

Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329
[18]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[19]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure & Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411
[20]

Friedemann Brock, Francesco Chiacchio, Giuseppina di Blasio. Optimal Szegö-Weinberger type inequalities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 367-383. doi: 10.3934/cpaa.2016.15.367

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences