For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by $$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<s<2}, \end{cases}$$ where $c_s$ is a normalizing constant, so that $g_s \ast (\cdot) = |\nabla|^{-2+s}(\cdot)$.
I've come across the following singular integral operator in my research: $$T_{s,v} f(x) := \int_{\mathbb{R}^2}K_{s,v}(x,y)f(y)dy, \qquad f\in C_c^\infty(\mathbb{R}^2)$$ $$K_{s,v}(x,y) := \nabla^{\otimes 2}g_s(x-y) : (v(x)-v(y))^{\otimes 2}, \qquad x\neq y,$$ where $v:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is a vector field, which we can assume to be as nice as we want it to be, and $:$ denotes the Frobenius inner product. By considerations of scaling, I would expect $T_v$ to be smoothing of order $2-s$. More precisely, I would expect the operator $$f\mapsto |\nabla|^{1-\frac{s}{2}}T_{s,v}(|\nabla|^{1-\frac{s}{2}}f), \qquad f\in C_c^\infty(\mathbb{R}^2)$$ to have an extension which is bounded on $L^2(\mathbb{R}^2)$.
When $s=0$ and therefore $g_s$ is the 2D Coulomb potential, the desired boundedness property follows from the $L^2$ boundedness of Calder'{o}n $d$-commutators proven by Christ and Journ'{e} Polynomial growth estimates for multilinear singular integral operators (see Section 5). Is there anything know in the literature for the Riesz case $0<s<2$? So far, I have not found what I'm looking for--I'm not a harmonic analyst, though--and I'm not exactly sure how to prove what I want either. Perhaps, one could try computing the kernel of $T_{s,v}$ using the principal value representation of the fractional Laplacian, but this seems quite messy.
