The Kato square root problem for divergence form elliptic operators with potential \(V : {\mathbb {R}}^{n} \rightarrow {\mathbb {C}}\) is the equivalence statement \(\left\| \left( L + V \right) ^{\frac{1}{2}} u \right\| _{2} \simeq \left\| \nabla u \right\| _{2} + \left\| V^{\frac{1}{2}} u \right\| _{2}\), where \(L + V := - \mathrm {div} (A \nabla ) + V\) and the perturbation A is an \(L^{\infty }\) complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying \(\left\| \left| V \right| ^{\frac{\alpha }{2}} u \right\| _{2} + \left\| (-\Delta )^{\frac{\alpha }{2}}u \right\| _{2} \lesssim \left\| \left( \left| V \right| - \Delta \right) ^{\frac{\alpha }{2}}u \right\| _{2}\) for all \(u \in D(\left| V \right| - \Delta )\) and some \(\alpha \in (1,2]\). The class of potentials that will satisfy such a condition is known to contain the reverse Hölder class \(RH_{2}\) and \(L^{\frac{n}{2}}\left( {\mathbb {R}}^{n} \right) \) in dimension \(n > 4\). To prove the Kato estimate with potential, a non-homogeneous version of the framework introduced by Axelsson, Keith and McIntosh for proving quadratic estimates is developed. In addition to applying this non-homogeneous framework to the scalar Kato problem with zero-order potential, it will also be applied to the Kato problem for systems of equations with zero-order potential.

中文翻译：

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带势散度形式算子的加藤平方根问题

具有势\（V：{\ mathbb {R}} ^ {n} \ rightarrow {\ mathbb {C}} \）的椭圆形算子的散度的Kato平方根问题是等价语句\（\ left \ | \ left （L + V \ right）^ {\ frac {1} {2}} u \ right \ | _ {2} \ simeq \ left \ | \ nabla u \ right \ | _ {2} + \ left \ | V ^ {\ frac {1} {2}} u \ right \ | _ {2} \），其中\（L + V：=-\ mathrm {div}（A \ nabla）+ V \）和扰动A是一个满足累加条件的\（L ^ {\ infty} \）复杂矩阵值函数。证明了这种关系适用于范围在某个正部门中且满足\（\ left \ | \ left | V \ right | ^ {\ frac {\ alpha} {2}} u \ right \ | _ {2} + \ left \ |（-\ Delta）^ {\ frac {\ alpha} {2}} u \ right \ | _ {2} \ lesssim \ left \ | \ left（\ left | V \ right |-\ Delta \ right）^ {\ frac {\ alpha} {2}} u所有\（u \ in D（\ left | V \ right |-\ Delta）\）和某些\（\ alpha \ in（1,2] \）的\ right \ | _ {2} \）。已知满足这种条件的电位包含反向荷尔德类别\（RH_ {2} \）和\（L ^ {\ frac {n} {2}} \ left（{\ mathbb {R}} ^ {n} \ right）\）尺寸\（n> 4 \）。为了有潜力证明Kato估计，开发了Axelsson，Keith和McIntosh引入的用于证明二次估计的框架的非均匀版本。除了将此非齐次框架应用于具有零阶电势的标量Kato问题之外，还将将其应用于具有零阶电势的方程组的Kato问题。