# American Institute of Mathematical Sciences

August  2018, 38(8): 4019-4040. doi: 10.3934/dcds.2018175

## On fractional Hardy inequalities in convex sets

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy 2 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France 3 Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Received  October 2017 Revised  February 2018 Published  May 2018

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodeckiĭ spaces of order $(s, p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<∞$ and zhongwenzy $0<s<1$, with a constant which is stable as $s$ goes to 1.

Citation: Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
##### References:
 [1] K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequality, Math. Nachr., 284 (2011), 629-638.  doi: 10.1002/mana.200810109. [2] L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621. [3] L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007. [4] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. [5] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [6] Z.-Q. Chen and R. Song, Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.  doi: 10.2748/tmj/1113247482. [7] E. Cinti and F. Ferrari, Geometric inequalities for fractional Laplace operators and applications, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1699-1714.  doi: 10.1007/s00030-015-0340-3. [8] E. B. Davies, A review of Hardy inequalities, The Maz'ya anniversary collection, Vol. 2 (Rostock, 1998), Oper. Theory Adv. Appl., Birkhäuser, Basel, 110 (1999), 55–67. doi: 10.1007/978-3-0348-8672-7_5. [9] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003. [10] B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. [11] B. Dyda and A. V. Vähäkangas, A framework for fractional Hardy inequalities, Ann. Acad. Sci. Fenn. Math., 39 (2014), 675-689.  doi: 10.5186/aasfm.2014.3943. [12] R. L. Frank, R. Seiringer, Sharp fractional Hardy inequalities in half-spaces, Around the Research of Vladimir Maz'ya. I, Int. Math. Ser. (N. Y. ), Springer, New York, 11 (2010), 161–167. doi: 10.1007/978-1-4419-1341-8_6. [13] R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015. [14] H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequal. Appl., 1 (1997), 25-46.  doi: 10.1155/S1025583497000039. [15] A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921. [16] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), 1-21.  doi: 10.1007/s00526-008-0173-6. [17] M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.  doi: 10.1016/j.jfa.2010.05.001. [18] V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955. [19] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329. [20] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow, 1990. [21] A. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.

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##### References:
 [1] K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequality, Math. Nachr., 284 (2011), 629-638.  doi: 10.1002/mana.200810109. [2] L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621. [3] L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007. [4] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. [5] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [6] Z.-Q. Chen and R. Song, Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.  doi: 10.2748/tmj/1113247482. [7] E. Cinti and F. Ferrari, Geometric inequalities for fractional Laplace operators and applications, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1699-1714.  doi: 10.1007/s00030-015-0340-3. [8] E. B. Davies, A review of Hardy inequalities, The Maz'ya anniversary collection, Vol. 2 (Rostock, 1998), Oper. Theory Adv. Appl., Birkhäuser, Basel, 110 (1999), 55–67. doi: 10.1007/978-3-0348-8672-7_5. [9] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003. [10] B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. [11] B. Dyda and A. V. Vähäkangas, A framework for fractional Hardy inequalities, Ann. Acad. Sci. Fenn. Math., 39 (2014), 675-689.  doi: 10.5186/aasfm.2014.3943. [12] R. L. Frank, R. Seiringer, Sharp fractional Hardy inequalities in half-spaces, Around the Research of Vladimir Maz'ya. I, Int. Math. Ser. (N. Y. ), Springer, New York, 11 (2010), 161–167. doi: 10.1007/978-1-4419-1341-8_6. [13] R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015. [14] H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequal. Appl., 1 (1997), 25-46.  doi: 10.1155/S1025583497000039. [15] A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921. [16] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), 1-21.  doi: 10.1007/s00526-008-0173-6. [17] M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.  doi: 10.1016/j.jfa.2010.05.001. [18] V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955. [19] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329. [20] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow, 1990. [21] A. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.
The set $\Sigma_\sigma(x)$ and the supporting hyperplane $\Pi_{x'}$
The distance of $y$ from $\partial K$ is smaller than its distance from the hyperplane
The set $K_x$ in the second part of the proof of Theorem 1.1
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