Fractional sublinear Sobolev inequality for $\mathcal{L}-$superharmonic functions
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| Format: | Preprint |
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2025
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| _version_ | 1866913940096679936 |
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| author | May, Aye Chan Seesanea, Adisak |
| author_facet | May, Aye Chan Seesanea, Adisak |
| contents | We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-αq},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+α}} \right\|_{L^{q,t}(\mathbb{R}^n \times \mathbb{R}^n)} \] 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 $(- Δ_{p})^α $ with $0 < α< 1$ and $1<p<2$. Related Gagliardo-Nirenberg interpolation for $\mathcal{L}$-superharmonic functions is also derived. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional sublinear Sobolev inequality for $\mathcal{L}-$superharmonic functions May, Aye Chan Seesanea, Adisak Analysis of PDEs 46E35, 31B35, 31B05, 35R11 We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-αq},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+α}} \right\|_{L^{q,t}(\mathbb{R}^n \times \mathbb{R}^n)} \] 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 $(- Δ_{p})^α $ with $0 < α< 1$ and $1<p<2$. Related Gagliardo-Nirenberg interpolation for $\mathcal{L}$-superharmonic functions is also derived. |
| title | Fractional sublinear Sobolev inequality for $\mathcal{L}-$superharmonic functions |
| topic | Analysis of PDEs 46E35, 31B35, 31B05, 35R11 |
| url | https://arxiv.org/abs/2507.10344 |