Extensions To The Fractional Chain Rule

In this talk, we establish a fractional chain rule in the context of weighted Triebel-Lizorkin spaces. This result notably extends the fractional chain rule to weighted Sobolev spaces with an integrability index less than one. Additionally, we determine an explicit relationship between the smoothness index, the integrability index, and the choice of weights. Furthermore, the fractional chain rule for smoothness index $0<s<1$ is extended to a normed fractional Fa\'a di Bruno inequality for $s>0$ within the framework of Sobolev spaces.