2020/07/31 by Lorenzo Brasco, David Gómez‐Castro, David Gómez-Castro +2 · 3 citations
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Algorithm #Combinatorics #Computer science #Equivalence (formal languages) #Function (biology) #Function space #Homogeneous #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Pure mathematics #Sobolev inequality #Sobolev space #Space (punctuation) #math.AP #math.FA
paper · pdf · doi:10.1007/s00526-021-01934-6
published in Calculus of Variations and Partial Differential Equations 60(2) (Springer Science+Business Media) · 31 pages, an error in the proof of Lemma A.1 has been fixed
openalex publication_date 2021/03/01 · arxiv created 2021/06/08 · arxiv updated 2022/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces \mathcal Ds,p (\mathbb Rn) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> and their embeddings, for s ∈ (0,1] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math> and p≥ 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> . They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For s p lt; n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mspace/><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math> or s = p = n = 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> we show that \mathcal Ds,p(\mathbb Rn) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is isomorphic to a suitable function space, whereas for s p ≥ n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mspace/><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math> it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.