Inequalities for operators

Ravi P. Agarwal (), Shusen Ding () and Craig Nolder ()
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Ravi P. Agarwal: Florida Institute of Technology, Department of Mathematical Sciences
Shusen Ding: Seattle University, Department of Mathematics
Craig Nolder: Florida State University, Department of Mathematics

Chapter Chapter 7 in Inequalities for Differential Forms, 2009, pp 225-321 from Springer

Abstract: Abstract 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.

Keywords: Real Number; Projection Operator; Maximal Operator; Convex Domain; Beltrami Operator (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/978-0-387-68417-8_7

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