Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter September 25, 2021

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

Quoc-Hung Nguyen, Yannick Sire and Juan-Luis Vázquez ORCID logo
From the journal Forum Mathematicum


This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

MSC 2010: 42B37; 35J25

Communicated by Christopher D. Sogge

Funding statement: The first author is supported by the Shanghai Tech University startup fund. The third author was partially funded by Project PGC2018-098440-B-I00 from MICINN, of the Spanish Government. Also, he partially performed as an honorary professor at University Complutense de Madrid.

A Appendix

In this appendix, we provide two results supporting our conjecture on the failure of the usual form of the Leibniz rule in the case of the restricted Laplacian. Our purpose is to relate a weighted (by a suitable power of the distance function) Lp-norm for p=1,2 of the function to the Lp-norm of its fractional Laplacian. We would like to make in particular three comments:

  1. By the very definition of the restricted Laplacian, since the functions are supported on Ω, the Leibniz rule reduces to estimate the integrals


    Since we are interested in estimating L2-norms in Ω, one is led to consider quantities of the type

  2. It is by now well-known that smooth functions that are compactly supported in Ω and have finite Hα semi-norm behave like dist(x,Ω)α close to the boundary of Ω.

  3. Finally, notice that there are two different ways to define a semi-norm (even in d) in W˙α,p(d), namely


    In the case of the whole space d and p=2, these latter norms are equivalent. Actually, according to [24], depending on p, these spaces are ordered for every α(0,1) in bounded domains and they are still equivalent for p=2.

According to the previous remarks, if one seeks for a counter-example, one would need to understand how the L2-norm in d of the commutator behaves with respect to its L2-norm in Ω. The following computations show that the boundary behavior plays a crucial role.

Let α(0,12). Let uεCc(B1(0)) be a cut-off function such that uε=1 in B1-2ε, uε=0 in B1-εc, and |uε|Cε, 0uε1. We easily get first

B1uε(x)2(1-|x|2)2α𝑑x1for all ε(0,110).

and for any 0<α<α0<12,




On the other hand, for any xB1,


Indeed, for xB1,




Moreover, for α<α1<α2<12,


where the second relation follows by (2.2). Combining this with (A.2) yields (A.1). As a consequence, we have



B1B1|uε(x)-uε(y)|2|x-y|d+2α𝑑x𝑑y0as ε0.

This is an explicit example that in a bounded domain for α<12 the Hardy inequality in L2 does not hold.

However, the analogous result in L1 does hold.

Theorem A.1.

Let uCc(Ω) be a solution to


in Ω, where Ω is a smooth bounded domain. Then


for any δ(0,α4)


Using Tε(u(x))=sign(u(x))min{ε,|u(x)|} as test function, we obtain


This implies




Letting ε0 yields




we have


Moreover, we can write


for any xd. Thus, by the standard regularity theory, we have


for any δ(0,α2). Since


where the second relation follows by (A.4), we get (A.3). ∎


[1] M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5725–5767. 10.3934/dcds.2015.35.5725Search in Google Scholar

[2] D. Brazke, A. Schikorra and Y. Sire, in preparation, 2020. Search in Google Scholar

[3] D. Brazke, A. Schikorra and Y. Sire, Characterization of bmo via Carleson measures on Riemannian manifolds, Int. Math. Res. Not. IMRN, to appear. Search in Google Scholar

[4] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lect. Notes Unione Mat. Ital. 20, Springer, Cham, 2016. 10.1007/978-3-319-28739-3Search in Google Scholar

[5] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), no. 5, 2052–2093. 10.1016/j.aim.2010.01.025Search in Google Scholar

[6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Search in Google Scholar

[7] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations 36 (2011), no. 8, 1353–1384. 10.1080/03605302.2011.562954Search in Google Scholar

[8] J. A. Carrillo, M. del Pino, A. Figalli, G. Mingione and J. L. Vázquez, Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, Lecture Notes in Math. 2186, Springer, Cham, 2017. 10.1007/978-3-319-61494-6Search in Google Scholar

[9] S. Y. A. Chang and R. A. Yang, On a class of non-local operators in conformal geometry, Chin. Ann. Math. Ser. B 38 (2017), no. 1, 215–234. 10.1007/s11401-016-1068-zSearch in Google Scholar

[10] Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal. 226 (2005), no. 1, 90–113. 10.1016/j.jfa.2005.05.004Search in Google Scholar

[11] P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 (2016), no. 2, Article ID 8. 10.1007/s40818-016-0017-1Search in Google Scholar

[12] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge University, Cambridge, 1990. Search in Google Scholar

[13] D. Frey, Paraproducts via H-functional calculus, Rev. Mat. Iberoam. 29 (2013), no. 2, 635–663. 10.4171/RMI/733Search in Google Scholar

[14] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. 10.1002/cpa.3160410704Search in Google Scholar

[15] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. 10.1002/cpa.3160460405Search in Google Scholar

[16] P. Kim, R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion, Trans. Amer. Math. Soc. 371 (2019), no. 6, 3917–3969. 10.1090/tran/7358Search in Google Scholar

[17] P. Kim, R. Song and Z. Vondraček, On the boundary theory of subordinate killed Lévy processes, Potential Anal. 53 (2020), no. 1, 131–181. 10.1007/s11118-019-09762-2Search in Google Scholar

[18] E. Lenzmann and A. Schikorra, Sharp commutator estimates via harmonic extensions, Nonlinear Anal. 193 (2020), Article ID 111375. 10.1016/ in Google Scholar

[19] D. Li, On Kato–Ponce and fractional Leibniz, Rev. Mat. Iberoam. 35 (2019), no. 1, 23–100. 10.4171/rmi/1049Search in Google Scholar

[20] D. Li and Y. Sire, Higher order Kato–Ponce estimates and counter-examples, in preparation. Search in Google Scholar

[21] R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Partial Differential Equations 39 (2014), no. 9, 1780–1790. 10.1080/03605302.2013.864304Search in Google Scholar

[22] Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 (2018), no. 10, 1502–1539. 10.1080/03605302.2018.1475492Search in Google Scholar

[23] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 4, 831–855. 10.1017/S0308210512001783Search in Google Scholar

[24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar

[25] P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122. 10.1080/03605301003735680Search in Google Scholar

[26] A. Zygmund, Trigonometric Series. Vol. I, II, 2nd ed., Cambridge University, New York, 1959. Search in Google Scholar

Received: 2020-08-17
Revised: 2021-08-23
Published Online: 2021-09-25
Published in Print: 2021-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston