Relating different conditions for the positivity of the Schrödinger operator (Q1801942)

Let \(L=-\Delta-(1/c)v\) be a Schrödinger operator in \(\mathbb{R}^ d\). The author shows that the Fefferman condition \[ \left({1\over {| Q|}} \int_ Q v^ p\right)^{1/p} \leq {c \over {\ell^ 2(Q)}},\qquad p>1,\;Q:\text{ cube}, \] where \(\ell(Q)\) is the side length of \(Q\), implies the Kerman-Sawyer condition \[ \int_ Q M^ 2_ 1(\chi_ Q v)\leq c\int_ Q v, \qquad \text{for all }Q, \] where \[ M_ 1(f)(x)=\sup_{{{x\in Q} \atop {Q\text{ any cube }}}} {1 \over {| Q|^{1-1/d}}} \int_ Q f(y)dy, \] which is a necessary and sufficient condition for the positivity of the operator \(L\). He also shows that the Chang, Wilson and Wolff condition \[ {1\over {| Q|}} \int_ Q \ell^ 2(Q)v(y)\log^{2+\varepsilon} (1+\ell^ 2(Q)v(y))dy \leq c \] implies the Kerman-Sawyer condition with \(\int_ Q M^ 2_ 1 (\chi_ Q v)\) replaced by \(\int_{\mathbb{R}^ d} M^ 2_ 1(\chi_ Q v)\). The author's proof is direct and neither \(A_ \infty\) weights nor the Haar-type decomposition is used.