Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 437-446

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES

Juha Lehrbäck

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; juhaleh 'at' maths.jyu.fi

Abstract. We establish necessary conditions for domains \Omega \subset Rⁿ which admit the pointwise (p,\beta)-Hardy inequality

|u(x)| \le C d_\Omega(x)^1-\beta/pM_{2d_\Omega(x),q} (|\nabla u|d_\Omega^\beta/p)(x), u \in C₀^\infty(\Omega),

where 1 < q < p, d_\Omega(x) = dist(x,\partial\Omega), and M_R,q is a maximal operator. In particular, the complement of such a domain must have, even locally, Hausdorff dimension strictly greater than n - p + \beta.

2000 Mathematics Subject Classification: Primary 46E35, 26D15, 28A78.

Key words: Pointwise Hardy inequality, maximal function, Hausdorff content, Hausdorff dimension, Minkowski dimension.

Reference to this article: J. Lehrbäck: Necessary conditions for weighted pointwise Hardy inequalities. Ann. Acad. Sci. Fenn. Math. 34 (2009), 437-446.

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