will be valid for all bounded singular integrals, the lower bounds require some suitable non-degeneracy, and here we work with the Riesz transforms $$R_jf(x) = \lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}\frac{x_j-y_j}{|x-y|^{d+1}}f(y){\,\mathrm{d}}y, \quad f\in L^2({\mathbb{R}}^d),\quad \ j=1,\cdots, d.$$ We show that $$\label{intro1}
    C_{T}(S_{2}(b_1,b_2)+T_{2}(b_1,b_2))  \leq {\big\|[b_2,[b_1,T]]\big\|_{L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)}}\leq  C_{\varepsilon}(S_{2+\varepsilon}(b_1,b_2)+T_{2+\varepsilon}(b_1,b_2)),$$ where the constant $C_{\varepsilon}$ tends to infinity as $\varepsilon$ tends to zero and the joint conditions $S_p,T_p$, with $0<p<\infty,$ imposed on the complex valued functions $b_1,b_2$ are defined by $$\label{intro2}
    S_p(b_1,b_2)  = \sup_{Q}\left(\frac{1}{{|Q|}}\int_Q{|b_1-\langle b_1\rangle_Q|}^p\right)^{1/p}\left(\frac{1}{{|Q|}}\int_Q{|b_2-\langle b_2\rangle_Q|}^p\right)^{1/p}$$ and $$\label{intro3}
    T_p(b_1,b_2)  = \sup_{Q}\left(\frac{1}{{|Q|}}\int_Q{|b_1-\langle b_1\rangle_Q|}^p{|b_2-\langle b_2\rangle_Q|}^p\right)^{1/p},\qquad b_1,b_2,b_1b_2\in L^p_{loc}({\mathbb{R}}^d).$$ Here the supremums are taken over all cubes. Whenever it is well understood which functions $b_1,b_2$ are in question, we refer to these conditions shortly as $T_p$ and $S_p.$

We show by example that the lower bound in does not improve to $S_{2+\varepsilon}(b_1,b_2)+T_{2+\varepsilon}(b_1,b_2)$ for any $\varepsilon>0$ – that is, the obtained upper bound is not necessary. This leads us to consider joint conditions involving Young functions that can be made strictly weaker than $S_{2+\varepsilon}+T_{2+\varepsilon}$ for all $\varepsilon >0.$ Hence, we prove the commutator upper bound with these updated conditions with a version of the sparse domination principle introduced in Lerner and Ombrosi [@LO19].

Basic notation
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We denote $A \lesssim B$, if $A \leq 