AI 中文总结
研究非自治二阶抛物型微分算子相关里斯变换的$\mathrm{L}^p$界，通过$\mathrm{L}^p$预解式界确定复系数下$1<p \leq2$的最大开区间，实系数时证明可外推到全范围，基于两种几何的时空非对角界论证。
AI 中文摘要
我们建立了关于与具有有界系数且依赖于所有变量可测的散度形式的非自治二阶抛物型微分算子相关的里斯变换的$\mathrm{L}^p$界的首个结果。对于复系数情形，通过$\mathrm{L}^p$预解式界确定指数$1<p \leq2$的最大开区间，该区间总是包含$2$的下抛物型索伯列夫共轭，且在空间维数$n \geq 2$时结果是尖锐的。对于实系数，我们证明可外推到全范围。我们的论证依赖于基于两种互补几何的新颖时空非对角界：小尺度上的抛物立方体和大尺度上以抛物贝塞尔势的半阶时间导数为模型的区域。
英文摘要
We establish the first results on $\mathrm{L}^p$ bounds for Riesz transforms associated with non-autonomous second order parabolic differential operators in divergence form with bounded coefficients that depend measurably on all variables. In the case of complex coefficients, we identify the maximal open range of exponents $1<p \leq2$ through the availability of $\mathrm{L}^p$ resolvent bounds. This open range always contains the lower parabolic Sobolev conjugate of $2$ and the result is sharp in spatial dimension $n \geq 2$. For real coefficients, we prove extrapolation to the full range. Our argument relies on novel space-time off-diagonal bounds based on two complementary geometries: parabolic cubes on small scales and regions modeled after the half-order time derivative of a parabolic Bessel potential on large scales.
CommentsSubmitted. Some typos eliminated. 36 pages, 4 figures. Comments are welcome
