Fractional sublinear Sobolev inequality for $\mathcal{L}-$superharmonic functions

We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions $$\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\frac{nq}{n-\alpha q},t}({\mathbb{R}}^n)}\le c{\parallel \frac{u(x)-u(y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\frac{n}{q}+\alpha }}\parallel }_{L^{q,t}({\mathbb{R}}^n\times {\mathbb{R}}^n)}$$ in the sublinear case $p-1<q<1$ and $p-1\le t\le \mathrm{\infty }$. The nonlocal nonlinear elliptic operator $\mathcal{L}$ is modeled from the fractional $p$-Laplacian $(-{\mathrm{\Delta }}_p{)}^{\alpha }$ with $0<\alpha <1$ and $1<p<2$. Related Gagliardo-Nirenberg interpolation for $\mathcal{L}$-superharmonic functions is also derived.