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Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae for fractional Sobolev spaces via interpolation and extrapolation

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Abstract

The real interpolation spaces between \(L^{p}({{\mathbb {R}}}^{n})\) and \({\dot{H}}^{t,p}({{\mathbb {R}}}^{n})\) (resp. \(H^{t,p}({{\mathbb {R}}}^{n})\)),   \(t>0,\) are characterized in terms of fractional moduli of smoothness, and the underlying seminorms are shown to be “the correct” fractional generalization of the classical Gagliardo seminorms. This is confirmed by the fact that, using the new spaces combined with interpolation and extrapolation methods, we are able to extend the Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae, as well as the Bourgain–Brezis–Mironescu convergence theorem, to fractional Sobolev spaces. On the other hand, we disprove a conjecture of Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) suggesting fractional convergence results given in terms of classical Gagliardo seminorms. We also solve a problem proposed in Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) concerning sharp forms of the fractional Sobolev embedding.

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Notes

  1. We assume that \(p>1,\) since, as in [5], our main focus will be the fractional Sobolev spaces defined via \((-\Delta )^{\frac{t}{2}}.\) The case \(p=1\) requires a separate treatment.

  2. See Sect. 4 for background information on interpolation and extrapolation.

  3. Throughout the paper, the symbol \(f \lesssim g\) indicates the existence of a constant \(c > 0\), which is independent of all essential parameters, such that \(f \le c \, g\). The symbol \(f \approx g\) means that \(f \lesssim g\) and \(g \lesssim f\).

  4. From this point of view, the constants that appear in (1.3) and (1.4) are connected to the computation of the underlying K-functionals.

  5. There is a method to define and compute the normalizations in order insure the continuity of the interpolation scales. In particular, the normalizations for other methods of interpolation are also known. For example, for the complex method of Calderón, \(c_{s}=1;\) for the \(J-\)method, \(c_{s,q} =\left( (1-s)sq^{\prime }\right) ^{-1/q^{\prime }},\) where \(\frac{1}{q} +\frac{1}{q^{\prime }}=1.\) In the case of quasi-Banach spaces some adjustments are necessary (cf. [16, p. 35] for more details).

  6. The proof of property (1.6) related to \((L^p({{\mathbb {R}}}^n), {\dot{W}}^k_p({{\mathbb {R}}}^n))\) can be seen, e.g., in [27, Proposition 3, p. 139].

  7. \({\dot{H}} ^{t,p}({\mathbb {R}}^{n})\) is also known as the space of Riesz potentials, while \(H^{t,p}({\mathbb {R}}^{n})\) is the space of Bessel potentials (cf. [27]).

  8. The concept of fractional differences was already used by Butzer et al. [8] to introduce fractional order moduli of smoothness, which has recently become a powerful tool in approximation theory, e.g., in the study of sharp inequalities between moduli of smoothness (Jackson–Marchaud–Ulyanov inequalities), cf. [21, 22, 29, 30] and the references therein.

  9. For the details of the computation we refer to [21], with [8, 33] as a forerunners.

  10. In this paper we shall be interested in the case \(1<p<\infty .\)

  11. Note also that (1.16) enables us to introduce Sobolev spaces of any order \(s>0\) (not necessarily \(s<1\) like in (2.1)), via higher order differences. For instance, one can introduce for \(s\in (0,2)\),

    $$\begin{aligned} \Vert f\Vert _{{\dot{W}}^{s,p}({{\mathbb {R}}}^{n}),2}=\left( \int _{{{\mathbb {R}}}^{n} }\int _{{{\mathbb {R}}}^{n}}\frac{|f(x+2h)-2f(x+h)+f(x)|^{p}}{|h|^{sp+N} }\,dx\,dh\right) ^{\frac{1}{p}}. \end{aligned}$$

  12. The appearence of the prefactor \((s(t-s)p)^{-\frac{1}{p}}\) in front of \(\Vert f\Vert _{L^p({{\mathbb {R}}}^n)}\) is dictated by standard normalizations in interpolation theory. This will become clear later, cf. Lemma 4.8.

  13. Spaces of periodic functions are defined similarly as their analogues on \({{\mathbb {R}}}^n\), simply replacing \(L^p({{\mathbb {R}}}^n)\) by \(L^p({{\mathbb {T}}})\).

  14. The limiting values \(p=1,\infty \) are also adimissible.

  15. The assumption that the couple \((A_{0},A_{1})\) is ordered is not restrictive, i.e., a corresponding embedding result still holds for general couples \((A_{0},A_{1})\). However, this embedding is simplified when dealing with ordered couples, which will be the only case of interest in this paper.

  16. Here we must take into account that \(F_{p,2}^{{\bar{r}} }({{\mathbb {R}}}^{n})\) can be identified with the vector-valued space \(L^{p}({{\mathbb {R}}}^{n};\ell _{2}^{{\bar{r}}}({{{\mathbb {N}}_0}}))\).

  17. In particular, this argument with \(t= \frac{n}{p} \in {{{\mathbb {N}}}}\) completes the proof of [20, Theorem 8].

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Acknowledgements

We are grateful to Petru Mironescu for his constant interest and encouragement during the preparation of this paper and to Sergey Tikhonov for precious information concerning fractional differences and moduli of smoothness. The authors would also like to thank the referee for his/her comments.

Funding

The first named author has been partially supported by the LABEX MILYON (ANR-10- LABX-0070) of Université deLyon,within the program “Investissement d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and by MTM2017-84058-P (AEI/FEDER, UE).

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  1. Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Oscar Domínguez

  2. Instituto Argentino de Matematica, Buenos Aires, Argentina

    Mario Milman

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  1. Oscar Domínguez
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Correspondence to Oscar Domínguez.

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Communicated by A. Mondino.

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Domínguez, O., Milman, M. Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae for fractional Sobolev spaces via interpolation and extrapolation. Calc. Var. 62, 43 (2023). https://doi.org/10.1007/s00526-022-02383-5

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