# Smoothing property of a certain singular integral operator of non-convolution type

For simplicity, suppose that the dimension $$d=2$$, and let $$g_s(x)$$ be the Coulomb or Riesz potential defined by $$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0 where $$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 $$0? So far, I have not found what I'm looking for--I'm not a harmonic analyst, though--and I'm not exactly sure how to prove what I want either. Perhaps, one could try computing the kernel of $$T_{s,v}$$ using the principal value representation of the fractional Laplacian, but this seems quite messy.

• Did you check the paper of L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93–140.? This is also called the Hörmander-Mikhlin Theorem. Since part of your examples are indeed homogeneous this theorem should apply. For the non-homogeneous examples I guess that some estimates could replace homogeneity. Jan 31, 2021 at 19:36
• @Bazin Thanks for your comment. I'm familiar with the HM multiplier theorem. Perhaps, I'm misunderstanding your suggestion, but I'm not sure why the HM theorem is relevant as my operator has a commutator structure, and therefore is not translation-invariant. Jan 31, 2021 at 19:55
• @Bazin I suppose one could try Taylor expanding $v(x)-v(y)$ to get at leading order that $T_{s,v}$ is essentially a composition of Riesz potential and a multiplication operator. Then one could use a Kato-Ponce type inequality to deal with the fractional derivatives. This seems sub-optimal, though, based on what I know for the Coulomb case, it would require more regularity on $v$ than Lipschitz, which is all that's needed in the aforementioned boundedness result for Calder\'{o}n $d$-commutators. Jan 31, 2021 at 21:32