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If Ω is a finite measure domain we show that several Poincaré, Hardy-type, or multiplicative inequalities as well as classical Sobolev embedding theorems on W 0 (Ω) may be extended to versions with singular or degenerate weights involving powers of the distance to the boundary function provided that ∂Ω is “fractal" in the sense that ∂Ω has interior Minkowski dimension  ̃ MD(∂Ω) < n . For unbounded non-finite measure domains such extensions may also often be made if ∂Ω satisfies a certain definition of “locally fractal". Mathematics subject classification (2000): Primary: 46E35, 26D10; Secondary: 35P05, 46B70, 54C25.

Some inequalities and embeddings for weighted W_0 spaces on domains with fractal boundaries