Radial extensions in fractional Sobolev spaces

@article{Brezis2018RadialEI,
  title={Radial extensions in fractional Sobolev spaces},
  author={Haim Brezis and Petru Mironescu and Itai Shafrir},
  journal={Revista de la Real Academia de Ciencias Exactas, F{\'i}sicas y Naturales. Serie A. Matem{\'a}ticas},
  year={2018},
  volume={113},
  pages={707-714}
}
  • H. Brezis, P. Mironescu, I. Shafrir
  • Published 1 March 2018
  • Mathematics
  • Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
Given $$f:\partial (-1,1)^n\rightarrow {{\mathbb {R}}}$$f:∂(-1,1)n→R, consider its radial extension $$Tf(X):=f(X/\Vert X\Vert _{\infty })$$Tf(X):=f(X/‖X‖∞), $$\forall \, X\in [-1,1]^n{\setminus }\{0\}$$∀X∈[-1,1]n\{0}. Brezis and Mironescu (RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95:121–143, 2001), stated the following auxiliary result (Lemma D.1). If $$0<s<1$$0<s<1, $$1< p<\infty $$1<p<∞ and $$n\ge 2$$n≥2 are such that $$1<sp<n$$1<sp<n, then $$f\mapsto Tf$$f↦Tf is a bounded… 

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