## Nonlinear Potential Theory and Weighted Sobolev Spaces, Issue 1736The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems. |

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### Contents

I | 1 |

III | 2 |

V | 3 |

VI | 6 |

VII | 7 |

VIII | 10 |

IX | 12 |

XI | 15 |

XXXVII | 92 |

XXXVIII | 97 |

XL | 99 |

XLI | 104 |

XLII | 108 |

XLIII | 110 |

XLIV | 113 |

XLV | 115 |

XII | 16 |

XIII | 17 |

XIV | 19 |

XV | 23 |

XVI | 25 |

XVII | 37 |

XVIII | 40 |

XIX | 41 |

XX | 44 |

XXI | 47 |

XXII | 49 |

XXIII | 53 |

XXIV | 54 |

XXV | 58 |

XXVI | 59 |

XXVII | 62 |

XXVIII | 69 |

XXIX | 70 |

XXXI | 75 |

XXXII | 77 |

XXXIV | 80 |

XXXV | 88 |

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### Common terms and phrases

Adams-Hedberg Ap condition Ap weight arbitrary assume balls Bj belongs Bessel capacity Borel measurable Borel set bounded Br(a Br(x capacitary measures Carlsson compact set compact subsets constant of w converges Corollary cube Q D. R. Adams define the weighted denoted depends domain Equations equivalent exists a constant exists a positive finite Fubini's theorem Hausdorff capacity Hausdorff measures Hedberg Holder's inequality holds implies isoperimetric inequality JRN JRN last inequality Lipschitz domain M+(RN Ma,Pn(x measurable function Muckenhoupt non-weighted nonnegative obtain open set open subset positive constant positive Radon measure potential theory proof of Theorem Proposition prove radius Remark Riesz capacity Riesz potential right-hand side satisfying second inequality Section Sobolev spaces Sobolev type inequality Stein 99 strong doubling property subset of RN Suppose Suslin set V. G. Maz'ya variational capacities VIII Vmu\wdx weighted Sobolev spaces

### Popular passages

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Page 168 - Weighted Sobolev spaces and pseudodifferential operators with smooth symbols, Trans. Amer. Math. Soc. 269 (1982), 91-109.

Page 170 - WIENER, N., The Dirichlet problem, J. Math, and Phys. 3 (1924), 127146.

Page 168 - imbedding theorems and the spectrum of a selfadjoint elliptic operator, Izv. Akad. Nauk SSSR, Ser. Mat. 37 (1973), 356-385

Page 169 - SOBOLEV, SL, On a boundary value problem for polyharmonic equations, Mat. Sb. (NS) 2

Page 164 - for an arbitrary open set fi, Trudy. Mat. Inst. Steklov., 131 (1974) 51-63