The Kato Square Root Problem follows from an extrapolation property of the Laplacian

Abstract

Abstract. On a domain Ω ⊆ R d we consider second order elliptic systems in divergence form with bounded complex coefficients, realized via a sesquilinear form with domain V ⊆ H 1 (Ω). Under very mild assumptions on Ω and V we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends an earlier result of M c Intosh [24] to non-smooth coefficients.

L -estimates for the square root of elliptic systems with mixed boundary conditions

“…The case of non-symmetric forms has a long history and became known as Kato square root problem, see for example [7,9,11]. Within our setup it has been settled in [31,32] by a non-trivial refinement of the first-order method of Axelsson-Keith-McIntosh proposed in their seminal paper [12] and their pioneering application to mixed boundary value problems in [13]. The author has to concede that it is somewhat unfortunate that [32] and [31] treat systems with lower-order terms only implicitly.…”

“…Within our setup it has been settled in [31,32] by a non-trivial refinement of the first-order method of Axelsson-Keith-McIntosh proposed in their seminal paper [12] and their pioneering application to mixed boundary value problems in [13]. The author has to concede that it is somewhat unfortunate that [32] and [31] treat systems with lower-order terms only implicitly. He sees this paper as the right moment to close this gap and shortly review the underlying methods in Section 3 to prove the following Theorem 1.1.…”

“…We survey the first-order formalism and its practicability to the operator L under our geometric assumptions developed in [31,32]. This leads to Theorem 1.1 and the statements of Theorem 1.5 when p = 2.…”

“…On the concrete level, the goal of [31] was to prove quadratic estimates for the particular choice of Π B under a set of hypotheses called (H1) -(H7). We do not need to recall them here and the interested reader can refer to [31,Sec. 5].…”

“…For the operators above, they have been verified in detail in [31, Sec. 6.1] with two exceptions: The accretivity assumption (H2), which is precisely (3.1), and the regularity assumption (H7), whose verification in [31] was subject to an additional assumption called Assumption (E) that became a true theorem only later on in [32,Thm. 4.4].…”