## Function Spaces and Potential TheoryFunction spaces, especially those spaces that have become known as Sobolev spaces, and their natural extensions, are now a central concept in analysis. In particular, they play a decisive role in the modem theory of partial differential equations (PDE). Potential theory, which grew out of the theory of the electrostatic or gravita tional potential, the Laplace equation, the Dirichlet problem, etc. , had a fundamen tal role in the development of functional analysis and the theory of Hilbert space. Later, potential theory was strongly influenced by functional analysis. More re cently, ideas from potential theory have enriched the theory of those more general function spaces that appear naturally in the study of nonlinear partial differential equations. This book is motivated by the latter development. The connection between potential theory and the theory of Hilbert spaces can be traced back to C. F. Gauss [181], who proved (with modem rigor supplied almost a century later by O. Frostman [158]) the existence of equilibrium potentials by minimizing a quadratic integral, the energy. This theme is pervasive in the work of such mathematicians as D. Hilbert, Ch. -J. de La Vallee Poussin, M. Riesz, O. Frostman, A. Beurling, and the connection was made particularly clear in the work of H. Cartan [97] in the 1940's. In the thesis of J. Deny [119], and in the subsequent work of J. Deny and J. L. |

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### Contents

I | 1 |

III | 2 |

IV | 3 |

V | 4 |

VII | 5 |

IX | 6 |

X | 8 |

XI | 9 |

LIV | 155 |

LV | 156 |

LVI | 158 |

LVII | 164 |

LVIII | 176 |

LIX | 180 |

LX | 185 |

LXI | 187 |

XII | 10 |

XIII | 11 |

XIV | 13 |

XV | 14 |

XVII | 16 |

XVIII | 17 |

XIX | 19 |

XX | 24 |

XXI | 30 |

XXII | 34 |

XXIII | 38 |

XXIV | 45 |

XXV | 48 |

XXVII | 53 |

XXIX | 58 |

XXX | 62 |

XXXI | 66 |

XXXII | 68 |

XXXIII | 72 |

XXXIV | 78 |

XXXV | 81 |

XXXVI | 85 |

XXXVIII | 92 |

XXXIX | 97 |

XL | 104 |

XLI | 108 |

XLII | 111 |

XLIII | 116 |

XLIV | 122 |

XLV | 125 |

XLVI | 129 |

XLVIII | 140 |

XLIX | 142 |

L | 146 |

LI | 148 |

LII | 150 |

LIII | 152 |

LXIII | 191 |

LXIV | 195 |

LXV | 199 |

LXVI | 203 |

LXVII | 208 |

LXVIII | 213 |

LXIX | 215 |

LXXI | 219 |

LXXII | 227 |

LXXIII | 231 |

LXXIV | 233 |

LXXVI | 239 |

LXXVII | 240 |

LXXVIII | 243 |

LXXIX | 245 |

LXXX | 248 |

LXXXI | 251 |

LXXXII | 257 |

LXXXIII | 263 |

LXXXIV | 266 |

LXXXV | 277 |

LXXXVI | 278 |

LXXXVIII | 281 |

XC | 293 |

XCI | 301 |

XCII | 302 |

XCIII | 305 |

XCV | 312 |

XCVI | 314 |

XCVII | 316 |

XCVIII | 318 |

XCIX | 324 |

C | 325 |

CI | 329 |

351 | |

CIII | 363 |

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### Common terms and phrases

Adams apply approximation arbitrary assume assumption ball Banach space belongs Besov Bessel bounded called capacitary capacity Chapter choose claim classical clearly closed compact condition consequence consider constant construct contained continuous converges Corollary cubes defined Definition denote depending differential distribution easily easy elements equations equivalent estimate exists extended fact finite follows formula function given gives hand implies independent inequality integral kernel Lebesgue Lemma linear Math Maz'ya means measure Moreover neighborhood nonlinear potential norm Note observe obtain open set operator p)-capacity polynomial positive potential potential theory problem proof of Theorem Proposition prove references Remark replaced restriction result Riesz satisfies Section sense sequence Sobolev solution spaces sufficient supp Suppose theory true write