\int_{{\mathbb{R}}} (|y x|^2\wedge 1) \,\nu(dx)\leq K\Big( {\mathbbm{1}}_{\{|y|\leq 1\}} |y|^\theta +{\mathbbm{1}}_{\{|y|>1\}}|y|^{\beta+\epsilon}\Big),$$ which shows since $f_t\in L^\theta({\mathbb{R}}_+)$ is bounded. We remark that for $\theta'<\theta$ it holds $$\limsup_{t\to \infty} \nu(x\!:|x|\geq t) t^{\theta}<\infty \quad \Longrightarrow \quad \limsup_{t\to \infty} \nu(x\!:|x|\geq t) t^{\theta'}<\infty.$$ On the other hand, the assumption $g^{(k)}\in L^{\theta '} ((\delta ,\infty ))$ is stronger than $g^{(k)}\in L^{\theta } ((\delta ,\infty ))$, which creates a certain balance between these two conditions. Finally, we note that the assumption (A-$\log$) will be used only for the case $\theta =1$ (resp. $\theta=p$) in part (i) (resp. part (iii)) of Theorem \[maintheorem\].

More importantly, the conditions of assumption (A) guarantee that the quantity $$\int_{-\infty}^{t-\epsilon} g^{(k)}(t-s) \,dL_s, \qquad \epsilon >0,$$ is well-defined (typically, the above integral is not well-defined for $\epsilon =0$). The latter is crucial for the proof of Theorem \[maintheorem\](i). We recall that the condition $p\geq 1$ is imposed in Theorem \[maintheorem\](iii). We think that this condition might not be necessary, but the results of [@SamBra] applied in our proofs require $p\geq 1$. However, when the index $\alpha$ further satisfies $\alpha>k$, then the stochastic process $F$ at is continuous and condition $p\geq 1$ is not needed in Theorem \[maintheorem\](iii).

The conditions $\alpha \in (0,k-1/p)$ and $p>\beta$ of 