Let \(\mu\) be a Radon measure on \(\mathbb R^d\) satisfying \(\mu(B(x,r))\leq C_0r^n\) \((x\in \operatorname {supp}\mu\) and \(r>0)\), where \(n\) is a fixed number with \(00, | u| +| v| \leq r} \{\int_{2^jr\leq| x-y| }| K(x+u,y+v)-K(x,y)| \,d\mu(x) +\int_{2^jr\leq| x-y| }| K(x+u,y+v)-K(x,y)| d\mu(y)\}0\) and any cube \(Q\) in \(\mathbb R^d\) (\(\rho\)-weak boundedness), and \(T_\varepsilon(1)\), \(T_\varepsilon^*(1)\in \text{BMO}_\rho(\mu)\) uniformly on \(\varepsilon>0\), for some \(\rho>1\). Here \(T_\varepsilon f(x)=\int_{| x-y| >\varepsilon}K(x,y)f(y)\,d\mu(y)\), and \(\text{BMO}_\rho(\mu)\) is the space of \(L_{\text{loc}}^1(\mu)\) functions satisfying \(\sup_Q \mu(\rho Q)^{-1}\int_Q| f-\mu(Q)^{-1}\int_Q f\,d\mu| \,d\mu<\infty\). (ii) Under the assumptions in (i) and the condition \(T(1)=T^*(1)=0\), \(T\) is bounded on Besov spaces \(\dot B_{pq}^0(\mu)\), \(1