In this work, we study Riesz transforms associated to the generalized Stokes operator $A$ given by

\begin{align*}

A u = f \quad \Leftrightarrow \quad \left\{ \begin{aligned} - \operatorname{div} (\mu \nabla u) + \nabla \phi &= f && \text{in } \mathbb{R}^d, \\ \operatorname{div}(u) &= 0 && \text{in } \mathbb{R}^d. \end{aligned} \right.

Besides ellipticity of $\mu$, we only assume that the coefficients are bounded and measurable. We show that the associated Riesz transform $\nabla A^{- 1/2}$ is bounded on $\mathrm{L}^2 (\mathbb{R}^d)$ which is an extension of the resolution of Kato's square root problem for elliptic operators in divergence form to generalized Stokes operators. In addition, we study lower and upper bounds of the Riesz transforms in $\mathrm{L}^p (\mathbb{R}^d)$, \textit{i.e.},

\begin{align*}

\| \nabla A^{- 1/2} u \|_{\mathrm{L}^p} \leq C \| u \|_{\mathrm{L}^p} \quad \text{as well as} \quad \| \nabla A^{- 1/2} u \|_{\mathrm{L}^p} \geq c \| u \|_{\mathrm{L}^p}

\end{align*}

and provide ranges of $p$ for which such an upper or lower bound hold in general. This part can be seen as an extension of results in the monograph of Auscher.

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This is joint research with Luca Haardt.