Fractional Besov trace/extension-type inequalities via the Caffarelli–Silvestre extension
| dc.contributor.author | Li, Pengtao | |
| dc.contributor.author | Hu, Rui | |
| dc.contributor.author | Zhai, Zhichun | |
| dc.date.accessioned | 2023-04-25T16:16:45Z | |
| dc.date.available | 2023-04-25T16:16:45Z | |
| dc.date.issued | 2022 | |
| dc.description.abstract | Let u(·, ·) be the Caffarelli–Silvestre extension of f . The first goal of this article is to establish the fractional trace-type inequalities involving the Caffarelli–Silvestre extension u(·, ·) of f . In doing so, firstly, we establish the fractional Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of ∇(x, t)u(x, t). Then, we prove the fractional anisotropic Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of ∂tu(x, t) or (−Δ)−γ /2u(x, t) only. Moreover, based on an estimate of the Fourier transform of the Caffarelli–Silvestre extension kernel and the sharp affine weighted L p Sobolev inequality, we prove that the H˙ −β/2(Rn) norm of f (x) can be controlled by the product of the weighted L p-affine energy and the weighted L p-norm of ∂tu(x, t). The second goal of this article is to characterize non-negative measures μ on Rn+1+ such that the embeddings u(·, ·)Lq0,p0 μ (Rn+1) f Λ˙ p,q β (Rn ) hold for some p0 and q0 depending on p and q which are classified in three different cases: (1) p = q ∈ (n/(n + β), 1]; (2) (p, q) ∈ (1, n/β) × (1,∞); (3) (p, q) ∈ (1, n/β) × {∞}. For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak-type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractional Besov capacity of open sets. | |
| dc.description.uri | https://library.macewan.ca/cgi-bin/SFX/url.pl/DSH | |
| dc.identifier.citation | Li, P., Hu, R. & Zhai, Z. Fractional Besov Trace/Extension-Type Inequalities via the Caffarelli–Silvestre Extension. Journal of Geometric Analysis 32, 236 (2022). https://doi.org/10.1007/s12220-022-00975-3 | |
| dc.identifier.doi | https://doi.org/10.1007/s12220-022-00975-3 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14078/3075 | |
| dc.language.iso | en | |
| dc.rights | All Rights Reserved | |
| dc.subject | Sobolev inequality | |
| dc.subject | Sobolev logarithmic inequality | |
| dc.subject | Hardy inequality | |
| dc.subject | affine energy | |
| dc.subject | Carleson embedding | |
| dc.subject | fractional Laplacian | |
| dc.title | Fractional Besov trace/extension-type inequalities via the Caffarelli–Silvestre extension | en |
| dc.type | Article |