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

Technical Summary: $L^p$ Bounds for Parabolic Riesz Transforms with Rough Coefficients

Problem Statement
The paper addresses the boundedness of Riesz transforms associated with non-autonomous, second-order parabolic differential operators in divergence form on $\mathbb{R}^{n+1}$. The operator is defined as $H = \partial_t - \text{div}_x(A(x,t)\nabla_x)$, where the coefficient matrix $A$ is bounded, measurable, and elliptic, depending on all space-time variables. The associated Riesz transform is defined as $R_H = D H^{-1/2}$, where $D = (\nabla_x, D_t^{1/2})$ is the parabolic gradient involving a fractional time derivative of order $1/2$.

While $L^2$ boundedness for such operators was established via the solution to the parabolic Kato problem, the extension to $L^p$ spaces for $p \neq 2$ remains a significant challenge. In the elliptic setting with rough coefficients, $L^p$ bounds are typically restricted to a sub-range of exponents $1 < p < 2$ due to the lack of smoothness, and duality arguments fail for $p > 2$. For parabolic operators, the situation is further complicated by the non-local nature of the half-order time derivative $D_t^{1/2}$ and the coupling of space and time variables by the measurable coefficients, which prevents the use of classical Fourier multiplier techniques that separate variables.

Methodology and Key Ideas
The authors develop a novel extrapolation framework that relies on two complementary geometries to handle the limited space-time decay of the operator kernels:

1. Small Scales: Parabolic cubes $\Delta_r(x,t) = Q_r(x) \times I_r(t)$, reflecting the local geometry of the operator.
2. Large Scales: Regions modeled after the level sets of the half-order time derivative of a parabolic Bessel potential. These are "time-stretched" annuli where the time dimension is stretched by a factor of $N^2$ relative to the spatial dimension.

The core technical strategy involves establishing space-time off-diagonal estimates for the family of operators $(\lambda D E_\lambda)_{\lambda > 0}$, where $E_\lambda = (1 + \lambda^2 H)^{-1}$ is the resolvent. Unlike previous approaches that relied on interpolating $L^2$ energy estimates with $L^p$ bounds, the authors derive genuine $L^p$ off-diagonal estimates. This is crucial because the non-local fractional derivative $D_t^{1/2}$ means that even the underlying $L^2$ estimates do not provide sufficient decay for $p$ below the lower parabolic Sobolev conjugate of 2 ($2^\star$) using standard interpolation.

The proof proceeds via a bootstrapping argument along parabolic Sobolev conjugates. By iteratively applying extrapolation criteria (specifically a two-scale version of the Blunck–Kunstmann criterion and a new criterion for operators with limited decay), the authors extend the range of boundedness from $p=2$ down to a critical exponent.

Key Contributions and Results

1. Identification of the Maximal Range: The paper establishes the first $L^p$ bounds for parabolic Riesz transforms with coefficients depending measurably on all variables. It identifies the maximal open range of exponents $1 < p \leq 2$ for which $R_H$ is bounded. This range is determined by the critical exponent $p_-(H)$, defined as the infimum of $p$ for which the resolvent family $(E_\lambda)$ is $L^p$-bounded.

   • Theorem 1.3: For complex coefficients, $R_H$ is bounded on $L^p$ for all $p \in (p_-(H), 2]$. The authors prove that $p_-(H) = q_-(H)$ (where $q_-(H)$ is the infimum for the gradient family) and that this value lies in the interval $[1, 2^\star)$, where $2^\star = \frac{2(n+2)}{n+4}$ is the lower parabolic Sobolev conjugate of 2.

2. Real Coefficients: In the case where the coefficients $A$ are real-valued, the authors prove that $p_-(H) = 1$. Consequently, the Riesz transform is bounded on the full range $L^p$ for all $1 < p \leq 2$. Furthermore, the spatial gradient component $\nabla_x H^{-1/2}$ is shown to be of weak type $(1,1)$.

3. Sharpness: The results are shown to be sharp within the class of parabolic operators for spatial dimensions $n \geq 2$. Using irregular weak solutions constructed by Mooney, the authors demonstrate that for any $\epsilon > 0$, there exist coefficients such that $p_-(H) \geq 2^\star - \epsilon$. This confirms that the lower bound of the extrapolation range cannot be improved to 1 in the general complex case without additional assumptions.

4. New Extrapolation Framework: The paper introduces a boundedness criterion (Theorem 5.1) tailored to operators with "limited space-time decay" that do not fit into standard spaces of homogeneous type. This framework allows for the iterative extension of boundedness from $L^2$ down to the critical exponent by restarting the extrapolation argument from any known bounded exponent $q$.

Significance and Claims
The paper claims to provide the first results on $L^p$ bounds for Riesz transforms associated with non-autonomous parabolic operators with merely measurable coefficients depending on all variables. The significance lies in overcoming the coupling of space and time variables and the non-locality of the time derivative, which previously obstructed the application of standard Calderón–Zygmund theory.

The authors emphasize that their geometric insight—combining parabolic cubes with time-stretched regions—allows for the recovery of sufficient off-diagonal decay to replace classical coordinate-wise arguments. The result is sharp in dimensions $n \geq 2$, identifying the precise limitation imposed by the roughness of the coefficients. For real coefficients, the result recovers the full range $1 < p \leq 2$, extending previous results that were limited to autonomous operators or required more regularity.

The paper concludes by noting that the techniques developed are expected to be applicable to other non-local operators, such as generalized Stokes operators, and lists several open problems, including the weak type $(1,1)$ boundedness for the full Riesz transform with real coefficients and the behavior in dimension $n=1$.