Geodesic
On a higher-order Hardy inequality
Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 113-121
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The Hardy inequality $\int_\Omega|u(x)|^pd(x)^{-p}\dd x\le c\int_\Omega|\nabla u(x)|^p\dd x$ with $d(x)=\operatorname{dist}(x,\partial\Omega)$ holds for $u\in C^\infty_0(\Omega)$ if $\Omega\subset\Bbb R^n$ is an open set with a sufficiently smooth boundary and if $1
The Hardy inequality $\int_\Omega|u(x)|^pd(x)^{-p}\dd x\le c\int_\Omega|\nabla u(x)|^p\dd x$ with $d(x)=\operatorname{dist}(x,\partial\Omega)$ holds for $u\in C^\infty_0(\Omega)$ if $\Omega\subset\Bbb R^n$ is an open set with a sufficiently smooth boundary and if $1$. P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p=1$.
DOI : 10.21136/MB.1999.126250
Classification : 26D10, 31C15, 42B25, 46E35
Keywords: Hardy inequality; capacity; maximal function; Sobolev space; $p$-thick set 
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Edmunds, David E.; Rákosník, Jiří. On a higher-order Hardy inequality. Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 113-121. doi: 10.21136/MB.1999.126250

[1] D. R. Adams L. I. Hedberg: Function spaces and potential theory. Springer, Berlin, 1996. | MR

[2] D. E. Edmunds H. Triebel: Function spaces, entropy numbers and differential operators. Cambridge Tracts in Mathematics, vol. 120, Cambridge University Press, Cambridge, 1996. | MR

[3] D. Gilbarg N. S. Trudinger: Elliptic partial differential equations of second order. (2nd ed.), Springer, Berlin, 1983. | MR

[4] P. Hajlasz: Pointwise Hardy inequalities. Proc. Amer. Math. Soc. 127 (1999), 417-423. | DOI | MR | Zbl

[5] J. Heinonen T. Kilpeläinen O. Martio: Nonlinear potential theory of degenerate elliptic equations. Oxford Science Publications, Clarendon. Press, Oxford, 1993. | MR

[6] J. Kinnunen O. Martio: Hardy's inequalities for Sobolev functions. Math. Res. Lett. 4 (1997), no. 4, 489-500. | DOI | MR

[7] J. L. Lewis: Uniformly fat sets. Trans. Amer. Math. Soc. 308 (1988), no. 1, 177-196. | DOI | MR | Zbl

[8] V. G. Maz'ya: Sobolev spaces. Springer, Berlin, 1985. | MR | Zbl

[9] P. Mikkonen: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci.Fenn.Ser. A.I. Math, Dissertationes 104 (1996), 1-71. | MR | Zbl

[10] B. Opic A. Kufner: Hardy-type inequalities. Pitman Research Notes in Math. Series 219, Longman Sci. &Tech., Harlow, 1990. | MR

[11] E. M. Stein: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N.J., 1970. | MR | Zbl

[12] A. Wannebo: Hardy inequalities. Proc. Amer. Math. Soc. 109 (1990), no. 1, 85-95. | DOI | MR | Zbl

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