## Hausdorff measures, capacities, and Sobolev spaces with weights |

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### Contents

Introduction | 5 |

Weighted Hausdorff measures | 11 |

Capacity and measure densities | 24 |

Copyright | |

1 other sections not shown

### Common terms and phrases

5.8 and Lemma Aoo class Ap constant Ap weights B(xj,rj balls B(x,r Banach space Bessel kernels Bessel spaces Borel measure bounds for capacity closed set constants cq convolution Corollary 3.11 cube defined by w(x dim(F dimw(E doubling condition doubling weight F n B(x F with constants Fubini theorem G Ap gauge hatW Hardy-Littlewood maximal function hatW is linearly HatW(E Holder inequality hPtw(x hPtW(x,t increasing on F infimum kernels and Riesz Lebesgue integrable Lebesgue measure Lebesgue sense Let p G linearly increasing gauge locally integrable function lower bound mapping measurable function nonempty subset norm oo a.e. oo and let open set ordinary Hausdorff dimension outer measure prove Riesz transforms singular integral operator Suppose that F Theorem 5.8 upper bounds w^vdy weighted capacity density weighted content densities weighted Hausdorff dimension weighted Hausdorff measure weighted Sobolev space write