Equivalence between Regularity Theorems and Heat Kernel Estimates for Higher Order Elliptic Operators and Systems under Divergence Form

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Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I (2009, methodology): “ately from reverse H¨older inequalities described in Theorem 9.8 and Sobolev embeddings. For any dimension and system (m≥ 1,n≥ 1), with continuous in O1+n or vmocoeﬃcients, this is explicitly done in [10]. Proof. Applying the local boundedness property to u− u1(x0) on Whitney balls yields the desire”
Erratum to: Perturbation and interpolation theorems for the H∞-calculus with applications to differential operators (2006, result): “Remark 9.12. If the matrix a is assumed to be Holder continuous then we can show (via Corollary 6.3) that A has an H ∞ -calculus in H −1 p (R n ). We also refer to [8], [ 7 ], and [6] for related results.”
Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains (2011, methodology): “KIM The results in this paper and in [16] imply several generalizations of the interior Hölder estimate for higher order elliptic equations with VMO coefficients proved in [2] Proposition 50.”
The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $ (2001, result): “Another one is (indirectly) in [11]: it is a second order system with bounded measurable coecients and (1.17) which does not satisfy a Caccioppoli inequality, whereas it is shown in [ 5 ] that the G arding inequality (1.13) governs Caccioppoli inequality.”