Endpoint estimates for commutators of singular integrals related to Schrödinger operators
Let \(V\) be a nonnegative potential that satisfies a reverse Hölder inequality with exponent \(d/2\), where \(d\geq 3\) is the dimension. The class of operators \(\mathcal K_L\) is defined to be the class of sublinear operators \(T\) that map the Hardy space \(H^1 _L(\mathbb R^d)\) associated with \(L\) to \(L^1(\mathbb R^d)\). Here \(H^1 _L\) consists of all functions \(f\in L^1(\mathbb R^d)\) for which \(M_Lf:=\sup_{t>0} |e^{-tL f}|\in L^1(\mathbb R^d)\). For example, by definition, the maximal operator \(M_Lf\) is a \(\mathcal K_L\)-class operator. Of particular importance are the \(L\)-Riesz transforms \(\partial-{x_j} L^{-1/2}\). More generally, the author considers a class of operators which are associated with the Schrödinger operator \(L:=-\Delta+V\) as follows. First one defines a size function \(\rho\) by means of \(\rho(x):=\sup\{r>0:r^{-d+2}\int_{B(x,r)} V(y)dy\leq 1\}\) where \(x\in \mathbb R^d\). An operator \(T\) is a \((\delta,L)\)-Schrödinger-Calderón-Zygmund operator if it is associated with a kernel \(K\) which satisfies the usual \(\delta\)-Hölder continuity condition, in the classical sense, and additionally the size condition \(|K(x,y)|\lesssim_N \frac{1}{|x-y|^d} \big(1+\frac{|x-y|}{\rho(x)}\big)^{-N}\), for every \(N>0\). All \(L\)-Schrödinger-Calderón-Zygmund operators are of \(\mathcal K_L\)-class. The main questions revolve around finding sufficient conditions on symbols \(b\) such that the commutators \([T,b]f:=T(bf)-bT(f)\) map \(H^1 _L\) to \(L^{1,\infty}\) for \(\mathcal K_L\)-class operators \(T\). The main result of the paper shows that if \(T\in\mathcal K_L\) is of weak type \((1,1)\) and \(b\in\mathrm{BMO}(\mathbb R^d)\), then \(T\) maps \(H^1 _L\) into \(L^{1,\infty}\). In particular, this is true for the \(L\)-Riesz transforms. The author concludes the result above by a very precise decomposition of the commutator under the assumptions described before. For example, if \(T\in\mathcal K_L\) is linear and \(b\in\mathrm{BMO}(\mathbb R^d)\) the author shows that there exist bounded bilinear operators \(\mathfrak R_T,\mathcal G:H^1 _L\times\mathrm{BMO}\to L^1\) such that \( [T,b]f=\mathfrak R_T(f,b)+T(\mathcal G(f,b))\) and the operator \(\mathcal G\) does not depend on \(T\). Besides these main results the author provides a systematic study of \(\mathcal K_L\)-class operators, as well as many additional technical results that provide relations between \(\mathrm{BMO}\), \(\mathrm{BMO}_L\) and \(H^1 _L\), and identifies the largest subspace \(\mathcal H^1 _{L,b}\) of \(H^1 _L\) such that \([T,b]:\mathcal H^1 _{L,b}\to L^1\) for all Schrödinger-Calderón-Zygmund operators \(T\). Here \(\mathcal H^1 _{L,b}\) is a certain atomic \(H^1 _L\) space where the atoms satisfy cancellation conditions which also involve the symbol \(b\), as in [\textit{C. Pérez}, J. Funct. Anal. 128, No. 1, 163--185 (1995; Zbl 0831.42010)].
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