$\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$

Summary · qwen2.5:32b

The article establishes $\mathrm{L}^p$ bounds for Riesz transforms associated with non-autonomous second order parabolic differential operators, specifically for $1<p \leq 2$, marking the first results in this area with rough coefficients that depend measurably on all variables. The study identifies an open range of exponents through $\mathrm{L}^p$ resolvent bounds and uses novel space-time off-diagonal bounds based on parabolic cubes for small scales and regions modeled after a parabolic Bessel potential's half-order time derivative for large scales.

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Excerpt

arXiv:2607.05181v1 Announce Type: new Abstract: 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.