On the $L^p$-to-$L^q$ boundedness and compactness of commutators along monomial curves

Abstract: For exponents $p,q\in (1,\infty),$ we study the $L^p$-to-$L^q$ boundedness and compactness of the commutator $[b,H_{\gamma}] = bH_{\gamma} - H_{\gamma}b,$ where $H_{\gamma}$ is the Hilbert transform along the monomial curve $\gamma$ and the function $b$ is complex valued. We obtain a sparse form domination of the commutator and show that $b\in \operatorname{BMO}^{\gamma,\alpha}$ is a sufficient condition for boundedness in a half-open range of exponents $q\in[p,p+p(\Xi));$ in doing so we obtain a genuinely fractional case $q>p$ and also give a new proof of the case $q=p,$ due to Bongers, Guo, J. Li and Wick. Involving the $L^p$ improving phenomena of single scale averages, we show that $b\in \operatorname{VMO}^{\gamma,\alpha}$ is a sufficient condition for compactness in the same range of exponents $q\in[p,p+p(\Xi))$ as in the case of boundedness. For $q<p,$ we show that $b\in \dot L^r$ is a sufficient condition for compactness; the argument is new and we use it to obtain a new short proof of the same implication for commutators of Calder\'on-Zygmund operators, recently due to Hyt\"{o}nen, K. Li, Tao and Yang. For necessity, and now restricting to the plane, we develop the approximate weak factorization argument to the extent that it now works for all monomial curves, thus removing the extra assumption present in our earlier work that the graph $\gamma(\mathbb{R})\subset\mathbb{R}^2$ intersects adjacent quadrants. As a consequence in the plane, we obtain that all of the above sufficiency conditions are necessary. We present several open problems.