论文标题
潜在依赖性Riesz对完全不规则度量的无限性转换
Unboundedness of potential dependent Riesz transforms for totally irregular measures
论文作者
论文摘要
我们证明，对于完全不规则的度量，$ \ mathbb {r}^{d} $带有$ d \ geq3 $，$（d-1）$ - dimensional riesz transform $ 4 \ nabla_ {1} \ Mathcal {e} _ {a}^{v}（x，y）f（y）f（y）\，dμ（y）$$适用于schrödingeroperator $ l_ {a} a} a}^{a}^{v}^{v} = - $ \ Mathcal {e} _ {a}^{v} $不在$ l^{2}（μ）$上限制。这是Conde-Alonso，Mourgoglou和Tolsa获得的最新结果，用于自由空间椭圆运算符，其连续系数$ a $ a $，因为它允许在反向Hölder类$ rh__ {d} $中存在电势$ v $。我们通过获得内核$ \ nabla_ {1} \ Mathcal {e} _ {a}^{v} $的新指数衰减估计以及Hölder规则性估计，以由电位的关键半径函数确定的本地量表。
We prove that, for totally irregular measures $μ$ on $\mathbb{R}^{d}$ with $d\geq3$, the $(d-1)$-dimensional Riesz transform $$ T_{A,μ}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \, d μ(y) $$ adapted to the Schrödinger operator $L_{A}^{V} = -\mathrm{div} A \nabla + V$ with fundamental solution $\mathcal{E}_{A}^{V}$ is not bounded on $L^{2}(μ)$. This generalises recent results obtained by Conde-Alonso, Mourgoglou and Tolsa for free-space elliptic operators with Hölder continuous coefficients $A$ since it allows for the presence of potentials $V$ in the reverse Hölder class $RH_{d}$. We achieve this by obtaining new exponential decay estimates for the kernel $\nabla_{1} \mathcal{E}_{A}^{V}$ as well as Hölder regularity estimates at local scales determined by the potential's critical radius function.
