In this paper we investigate commutativity of rings with unity satisfying any one of the properties: \[ \begin{aligned} &\lbrace 1- g(yx^{m}) \rbrace \ [yx^{m} - x^{r} f (yx^{m}) \ x^s, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&\lbrace 1- g(yx^{m}) \rbrace \ [x^{m} y - x^{r} f (yx^{m}) x^{s}, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&y^{t} [x,y^{n}] = g (x) [f (x), y] h (x)\ {\mathrm and} \ \ [x,y^{n}] \ y^{t} = g (x) [f (x), y] h (x) \end{aligned} \] for some $f(X)$ in $X^{2} {\mathbb Z}[X]$ and $g(X)$, $ h(X)$ in ${\mathbb Z} [X]$, where $m \ge 0$, $ r \ge 0$, $ s \ge 0$, $ n > 0$, $ t > 0$ are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements $x$ and $y$ for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize a number of commutativity theorems established recently.
In this work we prove some sharp weighted inequalities on spaces of homogeneous type for the higher order commutators of singular integrals introduced by R. Coifman, R. Rochberg and G. Weiss in Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. As a corollary, we obtain that these operators are bounded on $L^{p}(w)$ when $w$ belongs to the Muckenhoupt’s class $A_{p}$, $p>1$. In addition, as an important tool in order to get our main result, we prove a weighted Fefferman-Stein type inequality on spaces of homogeneous type, which we have not found previously in the literature.