Inequalities for operators

  • Ravi P. AgarwalEmail author
  • Shusen Ding
  • Craig Nolder


The purpose of this chapter is to present a series of the local and global estimates for some operators, including the homotopy operator T, the Laplace–Beltrami operator Δ = d d * + d * d, Green’s operator G, the gradient operator , the Hardy–Littlewood maximal operator, and the differential operator, which act on the space of harmonic forms defined in a domain in R n , and the compositions of some of these operators. We introduce the Hardy–Littlewood maximal operator M s and the sharp maximal operator \({\rm M}_s^\#\) applied to differential forms in Section 7.1. We develop some basic estimates for Green’s operator ∇ ◦ T and d◦ T in Section 7.2. We establish some L s -estimates and imbedding inequalities for the compositions of homotopy operator T and Green’s operator G in Section 7.3. In Section 7.4, we prove some Poincaré-type inequalities for T◦ G and G◦ T. In Section 7.5, we obtain Poincaré-type inequalities for the homotopy operator T. In Section 7.6, we study various estimates for the composition T◦ H. In Section 7.7, we provide the estimates for the compositions of three operators. Finally, in Section 7.8, we offer some norm comparison theorems for the maximal operators.


Real Number Projection Operator Maximal Operator Convex Domain Beltrami Operator 
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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA
  2. 2.Department of MathematicsSeattle UniversitySeattleUSA
  3. 3.Department of MathematicsFlorida State UniversityTallahasseeUSA

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