Volume 5, Issue 3, June 2017, Page: 57-67
Specification of Fractional Poincare’ Inequalities for the Sequence Measures Generalization
Abdelilah Kamal H. Sedeeg, Mathematics Department, Faculty of Education, Holy Quran and Islamic Sciences University, Khartoum, Sudan; Mathematics Department, Faculty of Sciences and Arts-Almikwah, Albaha University, Albaha, Saudi Arabia
Shawgy H. Abd Alla, Mathematics Department, Faculty of Sciences, Sudan University of Science and Technology, Khartoum, Sudan
Received: Oct. 13, 2016;       Accepted: Nov. 8, 2016;       Published: Jun. 12, 2017
DOI: 10.11648/j.ajam.20170503.11      View  2382      Downloads  150
The Fractional Poincare’ inequalities in Rn are endowed with a fairly general sequence measure. We show a control of L2 norm by a non–Local quantity. The assumption on the sequence measure is that it satisfies the classical Poincare’ inequality, with general results. We also verify quantity of the tightness at infinity provided by the control on the fractional derivative in terms of a sequence of a weight growing at infinity. The illustration goes to the generator of the Ornstein-Uhlenbeck semi group and some estimates of its powers.
Poincare Inequalities, Non-Local Inequalities, Fractional Powers, Sequence Measure
To cite this article
Abdelilah Kamal H. Sedeeg, Shawgy H. Abd Alla, Specification of Fractional Poincare’ Inequalities for the Sequence Measures Generalization, American Journal of Applied Mathematics. Vol. 5, No. 3, 2017, pp. 57-67. doi: 10.11648/j.ajam.20170503.11
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D. del Castillo-Negrete, B. Carreras, V. Lynch, Nondiffusive transport in plasma turbulence: a fractional diffusion approach, Phys. Rev. Lett.94 (2005) 065003.
L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. 171 (3) (2010)1903–1930.
C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal. 143(1998) 273–307.
C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations 31 (2006)1321–1348.
C. Mouhot, R.M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math.Pures Appl. (9) 87 (2007) 515–535.
P. Gressman, R. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Natl. Acad. Sci. USA 107(2010) 5744–5749.
G. Di Nunno, B. Øksendal, F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, Universitext, Springer-Verlag,Berlin, 2009.
W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, second edition, John Wiley & Sons Inc., New York, 1971.
A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata, Rend. Mat. Appl. (5) 18 (1959) 95–139.
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007) 67–112.
L. Caffarelli, A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, preprint, 2010.
C. Mouhot, E. Russ and Y. Sire. Fractional Poincaré inequalities for general measures J. Math. Pures Appl. 95 (2011) 72–84.
R. Adams, J. Fourier, Sobolev Spaces, second edition, Pure Appl. Math.(Amst.), vol. 140, Elsevier/Academic Press, Amsterdam, 2003.
D. Bakry, F. Barthe, P. Cattiaux, A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including thelog-concave case, Electron. Commun. Probab. 13 (2008) 60–66.
M. Kanai, Rough isometries and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37(1985) 391–413.
X. Chen, Feng-Y. Wang and J. Wang, Perturbations of functional inequalities for Lévy type Dirichlet forms, Forum Mathematicum (June 2014). Volume 27, Issue 6, ISSN 0933-7741, Pages 3477–3507.
X. Chen, J.Wang Intrinsic ultracontractivity of Feynman–Kac semigroups for symmetric jump processes, Journal of Functional Analysis(1 June 2016),Volume 270, Issue 11, Pages 4152–4195.
Wang J. A simple approach to functional inequalities for non-local Dirichlet forms. ESAIM Probab Stat, 2014, 18: 503–513.
Chen X, Wang J. Functional inequalities for nonlocal Dirichlet forms with finite range jumps or large jumps. Stochastic Process Appl, 2014, 124: 123–153.
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009) 1–141.
J.-D. Deuschel, D. W. Stroock, Hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic Ising models,J. Funct. Anal. 92 (1990) 30–48.
M. Ledoux, The Concentration of Measure Phenomenon, Amer. Math. Soc., 2001.
L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Theory Related Fields 118(2000) 427–438.
D. Chafaï, Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities, J. Math. Kyoto Univ. 44 (2004)325–363.
I. Gentil, C. Imbert, The Lévy–Fokker–Planck equation:Φ-entropies and convergence to equilibrium, Asymptot. Anal. 59 (2008) 125–138.
N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math.Wiss., Band 180, Springer-Verlag, New York, 1972, translatedfrom the Russian by A.P. Doohovskoy.
D. Bakry, M. Émery, Propaganda forΓ2, in: From Local Times to Global Geometry, Control and Physics, Coventry, 1984/85, in: Pitman Res.Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, 1986, pp. 39–46.
M. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959) 1–11.
P. Auscher, On Necessary and Sufficient Conditions for Lp Estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates, Mem. Amer. Math. Soc., vol. 186, Amer. Math.Soc., 2007.
P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, P. Tchamitchian, The solution of the Kato square root problem for second order ellipticoperators on Rn, Ann. of Math. (2) 156 (2002) 633–654.
P. Auscher, A. McIntosh, E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008) 192–248.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, second edition, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin/New York, 1981.
R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967) 1031–1060.
E. Stein, The characterization of functions arising as potentials I, Bull. Amer. Math. Soc. 67 (1961) 102–104.
T. Coulhon, E. Russ, V. Tardivel-Nachef, Sobolev algebras on Lie groups and Riemannian manifolds, Amer. J. Math. 123 (2001) 283–342.
E. Davies, One-Parameter Semigroups, second edition, London Math. Soc. Monogr. Ser., vol. 15, Academic Press Inc., London/New York, 1980.
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