Sparse gradient bounds for divergence form elliptic equations

Abstract

We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available. The linear results have the full range of weighted estimates with Muckenhoupt weights as a consequence.

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References in corpus (8)

• An elementary proof of the $A_2$ Bound
• On pointwise estimates involving sparse operators
• Sharp weighted norm estimates beyond Calderón-Zygmund theory
• A pointwise estimate for positive dyadic shifts and some applications
• On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators
• On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: Part II
• Domination of multilinear singular integrals by positive sparse forms
• Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems