An Introduction to Sobolev Spaces and Interpolation Spaces

By L. Tartar. 2007

810 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

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Citations in the corpus

Density Estimation in Infinite Dimensional Exponential Families (2013, background): “ H, and the spaces R(Cβ) and R(C⌊β⌋),R(C⌈β⌉) β−⌊β⌋,2 have equivalent norms. To prove Proposition B.3, we need the following result which we quote from Steinwart and Scovel (2012, Lemma 6.3) (also see Tartar, 2007, Lemma 23.1) that interpolates L2-spaces whose underlying measures are absolutely conti”
Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples (2014, neutral): “We finish this introduction by a few words on the history of interpolation (and see [4,5,23,24]).”
Fractional div-curl quantities and applications to nonlocal geometric equations (2017, background): “Lp(V1 odR n) kdsgkLp′(V1 odR n) +Λ , for a uniform constant C>0 depending only on s, p, and the dimension n. For weak Lp-spaces L(p,∞) and more generally Lorentz spaces L(p,q) we refer, e.g., to [18, 49]. Proof. This follows from a standard argument, we only give a sketch of the proof: by Theore”
Sobolev Spaces on Non-Lipschitz Subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{doc (2016, positive): “To prove (iv) for s = 1 one uses instead of (30) the bound (cf. [55, Lemma 17.”
A multigrid algorithm for the $p$-version of the Virtual Element Method (2017, background): “33 a one dimensional hp inverse inequality, the interpolation estimates [60, 61] and (3.”
Characterization of interpolation between Grand, small or classical Lebesgue spaces (2017, background): “Our definition of the interpolation space is different from the usual one (see [5, 22]) since we restrict the norms on the interval (0, 1).”
Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension (2011, background): “More precisely in [11] the author takes W s,p = B p,p, where B s p,q is the Besov-Lipschitz space.”
Local Profiles for Elliptic Problems at Different Scales: Defects in, and Interfaces between Periodic Structures (2015, background): “In (21), we have of course denoted by L the weak-L space, also called the Marcinkiewitz space (see [6, Chapter 1], [35, Chapter 1, Section 8], and also [34]”
Regularity of n/2-harmonic maps into spheres (2010, neutral): “We will introduce the pseudo-norm [v]D,s, a quantity which for s ∈ (0, 1) actually is equivalent to the local, homogeneous H-norm, see [Tar07], [Tay96].”
Regularity of minimizers of autonomous convex variational integrals (2013, methodology): “We refer to sections 30–32 in [28] for a proof of this theorem.”