Inequalities for Differential Forms

In this first chapter, we discuss various versions of the Hardy–Littlewood inequality for differential forms, including the local cases, the global cases, one weight cases, and two-weight cases. We know that differential forms are generalizations of the functions, which have been widely used in many fields, including potential theory, partial differential equations, quasiconformal mappings, nonlinear analysis, electromagnetism, and control theory; see [1–19], for example. During recent years new interest has developed in the study of the L ^p theory of differential forms on manifolds [20, 21]. For p = 2, the L ^p theory has been well studied. However, in the case of p ≠ 2, the L ^p theory is yet to be fully developed. The development of the L ^p theory of differential forms makes it possible to transport all notations of differential calculus in R ⁿ to the field of differential forms. The outline of this chapter is first to provide background materials, such as the definitions of differential forms and A-harmonic equations, some classes of weight functions and domains, and then, introduce different versions of Hardy–Littlewood inequalities on various domains with some specific weights or norms.

In the previous chapter, we have discussed various versions of the Hardy–Littlewood inequality for a pair of solutions u and v of the conjugate A-harmonic equation. The purpose of this chapter is to present some norm comparison inequalities for differential forms satisfying the conjugate A-harmonic equations, which have been recently established in [104]. Since the proofs display a general method to obtain L ^p -estimates, we include most of them in this chapter. Also, we always assume that 1 < p < ∞ and p ⁻¹ + q ⁻¹ = 1 throughout this chapter.

We begin this chapter with the following weak reverse Hölder inequality, which is due to C. Nolder [71] and will be used repeatedly later.

In Chapter 3, we have discussed various versions of the Poincaré-type inequalities in which we have estimated the norm of u – u _B in terms of the corresponding norm of du. In this chapter, we develop a series of estimates which provide upper bounds for the norms of ≰ u (if u is a function) or du (if u is a form) in terms of the corresponding norm u – c, where c is any closed form. These kinds of estimates are called the Caccioppoli-type estimates or the Caccioppoli inequalities. In Section 4.2, we study the local A _r (Ω)-weighted Caccioppoli inequalities. The local Caccioppoli inequalities with two-weights are discussed in Section 4.3. The global versions of Caccioppoli inequalities on Riemannian manifolds and bounded domains are developed in Sections 4.4 and 4.5, respectively. In Section 4.6, we present Caccioppoli inequalities with Orlicz norms. Finally, in Section 4.7, we address few versions of Caccioppoli inequalities related to the codifferential operator d ^* .

In recent years various versions of imbedding theorems for differential forms have been established. The imbedding theorems for functions can be found in almost every book on partial differential equations, see Sections 7.7 and 7.8 in [274], for example. For different versions of imbedding theorems, see [20, 275–280, 32, 268, 81]. Many results for Sobolev functions have been extended to differential forms in R ⁿ . The imbedding theorems play crucial role in generalizing the theory of Sobolev functions into the theory of differential forms. The objective of this chapter is to discuss several other versions of imbedding theorems for differential forms. We also explore some imbedding theorems related to operators, such as the homotopy operator T and Green’s operator G. We first study the imbedding theorems for quasiconformal mappings in Section 5.2. Then, we establish an imbedding theorem for differential forms satisfying the nonhomogeneous A-harmonic equation in Section 5.3. In Section 5.4, we present the A _r (Ω)-weighted imbedding theorems related to the gradient operator and the homotopy operator. In Section 5.5, we explore some A ^r (λ, Ω)-weighted imbedding theorems and \(A_{r}^{\lambda} \)(Ω)-weighted imbedding theorems. In Section 5.6, we develop some L ^s -estimates and imbedding theorems for the compositions of operators. Finally, in Section 5.7, we study the two-weight imbedding inequalities.

In this chapter, we will present various versions of the reverse Hölder inequality which serve as powerful tools in mathematical analysis. The original study of the reverse Hölder inequality can be traced back in Muckenhoupt–s work in [145]. During recent years, different versions of the reverse Hölder inequality have been established for different classes of functions, such as eigenfunctions of linear second-order elliptic operators [281], functions with discrete-time variable [282], and continuous exponential martingales [119].

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.

In this chapter, we first present some integral inequalities related to a strictly increasing convex function on [0, ∞) and then improve the Hölder inequality with L ^p (log L) ^α (Ω)-norms to the case 0 < p, q < ∞. Next, we prove L ^p (log L) ^α (Ω)-integrability of Jacobians. We also show that the integrability exponents described in Theorem 8.2.7 are the best possible.