As an operator of order \( 0<2s<2 \), the fractional Laplacian plays a crucial role in connecting the identity operator to the classical Laplacian. To better understand this transition, a first-order expansion in \( s \) has recently been developed, offering new insights into this mathematical interplay.

Analogously to the linear case, the fractional \( p \)-Laplacian can be viewed as an interpolating operator between the identity and the classical \( p \)-Laplacian. A key element in the first-order expansion at \( s=0 \) is an operator of logarithmic type, which we refer to as the logarithmic \( p \)-Laplacian.

In this talk, I will explore this operator and discuss the specific challenges that arise when moving from a linear to a nonlinear framework. In particular, we will examine its variational structure and the asymptotic behaviour of the principal eigenvalue and eigenfunction as \( s \) tends to zero. Finally, we will derive a Faber-Krahn-type inequality for the principal eigenvalue of the logarithmic \( p \)-Laplacian and establish a Hardy-type inequality for the associated function spaces.

This talk is based on joint work with B. Dyda (Wrocław) and F. Sk (Chennai).