Regularity of superposition operators of mixed fractional order
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917497145393152 |
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| author | Bhowmick, Souvik Ghosh, Sekhar Kumar, Vishvesh Lakshmi, R. |
| author_facet | Bhowmick, Souvik Ghosh, Sekhar Kumar, Vishvesh Lakshmi, R. |
| contents | We extend the De Giorgi--Nash--Moser theory to superposition operators of mixed fractional operators. In particular, we investigate several regularity properties for this class of operators. We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local Hölder continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions.
Our results apply to both fixed-sign and sign-changing solutions involving mixed local--nonlocal superposition fractional operators. Notably, the results are new even in the classical linear case $p=2$, demonstrating the broader applicability of the techniques developed in this work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15346 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularity of superposition operators of mixed fractional order Bhowmick, Souvik Ghosh, Sekhar Kumar, Vishvesh Lakshmi, R. Analysis of PDEs 35B65, 35D30, 35B45, 35R09, 35R11, 35M12 We extend the De Giorgi--Nash--Moser theory to superposition operators of mixed fractional operators. In particular, we investigate several regularity properties for this class of operators. We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local Hölder continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions. Our results apply to both fixed-sign and sign-changing solutions involving mixed local--nonlocal superposition fractional operators. Notably, the results are new even in the classical linear case $p=2$, demonstrating the broader applicability of the techniques developed in this work. |
| title | Regularity of superposition operators of mixed fractional order |
| topic | Analysis of PDEs 35B65, 35D30, 35B45, 35R09, 35R11, 35M12 |
| url | https://arxiv.org/abs/2605.15346 |