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Towards a Unified Theory of Sobolev Inequalities

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Special Functions, Partial Differential Equations, and Harmonic Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 108))

Summary

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated with a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.

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Notes

  1. 1.

    This means that if f n  ≥ 0, and f n ↑ f, then \(\left \Vert f_{n}\right \Vert _{X} \uparrow \left \Vert f\right \Vert _{X}\) (i.e., Fatou’s Lemma holds in the X norm).

  2. 2.

    actually using the available cancellation, we have \(2f(y) =\int _{ -\infty }^{y}f^{{\prime}}(s)ds -\int _{y}^{\infty }f^{{\prime}}(s)ds\), therefore the constant of the embedding can be improved,

    $$\displaystyle{\left \Vert f\right \Vert _{\infty }\leq \frac{1} {2}\left \Vert f^{{\prime}}\right \Vert _{ 1}.}$$
  3. 3.

    For a detailed presentation of Maz’ya’s remarkable early work we refer to [51].

  4. 4.

    This is a somewhat disappointing turn of events for the developers of general abstract theories studying limiting inequalities (e.g., [56]) but our current understanding of Sobolev inequalities shows that: (a) Sobolev inequalities self-improve (cf. [5, 48, 79, 86]) and (b) the extrapolations of Sobolev inequalities take the form: “one inequality” implies a family of inequalities and in some cases “one inequality implies all”! (cf. [5, 29, 79]). We also refer to the forthcoming [84] for a connection with extrapolation of Sobolev inequalities à la Rubio de Francia.

  5. 5.

    For a detailed discussion on equivalences between different Lorentz *norms* we refer to [11, 57, 73].

  6. 6.

    The proof to deal with the case p = 1 is slightly different and hinges on a variant of (4), namely (cf. [86])

    $$\displaystyle{\int _{0}^{t}(f^{{\ast}{\ast}}(s) - f^{{\ast}}(s))s^{1/n}\frac{ds} {s} \leq c\int _{0}^{t}\left \vert \nabla f\right \vert ^{{\ast}}(s)ds.}$$
  7. 7.

    Actually (6) also makes sense, and gives sharp results, when p > n (cf. [80, Chapter 9]).

  8. 8.

    The isoperimetric profile was introduced by Maz’ya in the sixties and further developed by him in a number of publications (cf. [88] and the references therein). Independently, this useful tool was developed in parallel by geometers (cf. [12] and the references therein) and probabilists (cf. [67] and the references therein), and as we shall see plays an important role in our work formulating Sobolev pointwise inequalities on rearrangements.

  9. 9.

    In the general metric case it becomes the co-area inequality (cf. [18]).

  10. 10.

    In this case we have

    $$\displaystyle\begin{array}{rcl} \int _{0}^{\infty }I\left (\gamma (\{\left \vert f\right \vert > t\})\right )dt& \succeq & \int _{ 0}^{1/2}f^{{\ast}}(t)d\left (t\left (\log \frac{1} {t} \right )^{1/2}\right ) {}\\ & & \simeq \int _{0}^{1/2}f^{{\ast}}(t)\left (\log \frac{1} {t} \right )^{1/2}dt. {}\\ \end{array}$$
  11. 11.

    Here again we have to allow for *generalized* Lorentz spaces since the Gaussian profile although concave is not increasing. Indeed, I(t) is symmetric about 1∕2. Also note that the inequality

    $$\displaystyle{\int _{0}^{\infty }I\left (\gamma (\{\left \vert f\right \vert > t\})\right )dt \leq \left \|\nabla f\right \|_{ L^{1}(R^{n},\gamma )}}$$

    holds for functions f that do not vanish at the boundary. For example, for f = 1, the right hand is zero and

    $$\displaystyle\begin{array}{rcl} \int _{0}^{\infty }I\left (\gamma (\{\left \vert f\right \vert > t\})\right )dt& =& \int _{ 0}^{1}I\left (1)\right )dt {}\\ & =& \int _{0}^{1}0dt {}\\ & =& 0. {}\\ \end{array}$$
  12. 12.

    It is well known that the sets that realize the isoperimetric inequality are always hyperspaces: i.e., all but one of the variables are free.

  13. 13.

    For all Lipschitz function f on Ω, the modulus of the gradient is defined by

    $$\displaystyle{\vert \nabla f(x)\vert =\limsup _{d(x,y)\rightarrow 0}\frac{\vert f(x) - f(y)\vert } {d(x,y)}.}$$
  14. 14.

    The strong connection between the co-area formula and Sobolev embeddings had already been emphasized by Maz’ya in his pioneering fundamental work in the early sixties (cf. [88]).

  15. 15.

    See [52]. Using an approximation argument developed by E. Milman [92, Remark 3.3] it is possible to prove the main inequalities of this paper without this assumption (cf. [79] and the forthcoming [85]).

  16. 16.

    Under the assumption that f is Lip the calculations below can be justified (cf. [69, 75, 78])

  17. 17.

    In particular we require that for \(f \in Lip_{0}(\varOmega )\) we have \(f^{{\ast}}(\mu (\varOmega )^{-}) = 0\).

  18. 18.

    We refer to E. Milman’s papers for an account of the history of the problem. Emanuel, who belongs to the Milman family of mathematicians that includes David (grandfather), Vitali (father), Pierre (uncle) (cf. [54]), has no direct relation to Mario Milman.

  19. 19.

    An isoperimetric estimator is a continous concave function I: [0, 1] → [0, ∞], with I(0) = 0, increasing on (0, 1/2), symmetric about the point 1/2, and such that for t ∈ (0, 1∕2), 

    $$\displaystyle{I(t) \geq I_{(\varOmega,d,\mu )}(t).}$$
  20. 20.

    Independently, and in parallel, A. Calderón and his student Oklander defined and studied the K−functional, and real interpolation, e.g. in Oklander’s thesis at the University of Chicago (cf. [98]).

  21. 21.

    In other words we assume that (34) holds for X = L1 (Ω), and with \(\frac{t} {G(t)}\) replacing \(\frac{t} {I_{\varOmega }(t)}.\)

  22. 22.

    The iteration of Sobolev inequalities is not a new idea (cf. [88, Corollary 6.9/1 in page 379 and Theorem 7.6.5 in page 430]), the novelty here, if any, is the iteration of pointwise rearrangement inequalities

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Acknowledgements

We are grateful to E. Milman for a number of useful comments that helped improve the presentation.

The author J. Martín was Partially supported in part by Grants MTM2010-14946, MTM-2010-16232.

The author M. Milman was partially supported by a grant from the Simons Foundation (#207929 to Mario Milman).

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Authors and Affiliations

  1. Department of Mathematics, Universitat Autònoma de Barcelona, Barcelona, Spain

    Joaquim Martín

  2. Department of Mathematics, Florida Atlantic University, Boca Raton, FL, USA

    Mario Milman

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  1. Joaquim Martín
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  2. Mario Milman
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Correspondence to Mario Milman .

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  1. DePaul University, Chicago, Illinois, USA

    Constantine Georgakis

  2. Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  3. Roosevelt University, Chicago, Illinois, USA

    Wilfredo Urbina

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Martín, J., Milman, M. (2014). Towards a Unified Theory of Sobolev Inequalities. In: Georgakis, C., Stokolos, A., Urbina, W. (eds) Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-319-10545-1_13

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