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

TL;DR

This paper proves a new Sobolev-type inequality in Lorentz spaces for -superharmonic functions related to fractional p-Laplacian operators, extending classical inequalities to a nonlocal, nonlinear setting.

Contribution

It establishes a fractional sublinear Sobolev inequality and related Gagliardo-Nirenberg interpolation for -superharmonic functions, advancing analysis of nonlocal nonlinear elliptic operators.

Findings

Proved a Sobolev-type inequality in Lorentz spaces for -superharmonic functions.

Derived Gagliardo-Nirenberg interpolation inequalities for these functions.

Extended classical inequalities to the nonlocal, nonlinear fractional p-Laplacian context.

Abstract

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