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297. Maz'ya, V. G. (Ma3bH, B. r.): O perynHpHocTH Ha rpaHHne pemeHHH annHn­

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298. Maz'ya, V. G. (Ma3bH, B. r.): K TeopHH MHorOMepHoro onepaTOpa IIIpe.n;HHrepa (On the theory ofthe n-dimensional Schrodinger operator), Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964) 1145-1172.

299. Maz'ya, V. G. (Ma3bH, B. r.): ITOJIHrapMOHHl.IeCKaH eMKOCTb B TeopHH nepBoH KpaeBoH 3a.n;al.lH (Polyharmonic capacity in the theory ofthe first boundary value problem), Sibirsk. Mat. Zh. 6 (1965) 127-148.

300. Maz'ya, V. G. (Ma3bH, B. r.): O nOBe,!{eHHH B6nH3H rpaHHUhI pemeHHH 3a.n;al.lH LIHpHxJIe .n;nH aJIJIHllTHl.IeCKOrO ypaaHeHHH BToporo nOpH.n;Ka B ,!{H­BepreHTHoH ljlopMe, Mat. Zametki 2 (1967) 209-220. English translation: Behavior, near the boundary, of solutions of the Dirichlet problem for a second order elliptic equation in divergent form, Math. Notes 2 (1967) 610-617.

301. Maz'ya, V. G. (Ma3bH, B. r.): KJIaCChI MHo»<eCTB H Mep, CBH3aHHhIe C TeO­peMaMH BJIo»<eHHH (Classes of sets and measures connected with imbedding the­orems), in Teope.Mu aJl.o:J/Ce'H'U.H U ux npu.n.oJICe'H'U.H (Imbedding Theorems and Their Applications), Proc., Baku 1966 (L. D. Kudryavtsev (JI. LI. Ky.n;pHBueB), ed.), 142-159, Nauka, Moscow, 1970.

302. Maz'ya, V. G. (Ma3bH, B. r.): O HenpephIBHocTH B rpaHHl.IHolt TOl.lKe pe­meHHlt KBa3HJIHHeltHhIx aJIJIHnTHl.IeCKHX ypaBHeHHH, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 25:13 (1970) 42-55. Correction, ibid. 27:1 (1972) 160. Eng­lish translation: On the continuity at a boundary point of solutions of quasilinear equations, Vestnik Leningrad Univ. Math. 3 (1976) 225-242.

303. Maz'ya, V. G. (Ma3hH, B. r.): O HeKOTophIX HHTerpaJIhHhIX HepaBeHCTBax .n;JIH IjlYHKUHH MHorHX nepeMeHHhIX, Problemy Matematicheskogo Analiza, Leningrad. Univ. 3 (1972) 33-68. English translation: On certain integral inequalities for functions of many variables, J. Soviet Math. 1 (1973) 205-234.

304. Maz'ya, V. G. (Ma3bH, B. r.): 06 YCTpaHHMhIX oco6eHHocTHX OrpaHHl.IeHHhIX pemeHHlt KBa3HJIHHeHHhIx aJIJIHnTHl.IeCKHX ypaBHeHHlt JIIo6oro nOpH.n;Ka,

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Index

Aoo condition 83 Abel, N. H. 303 absolute continuity on Iines 238 Adams, D. R. 50,51,79-82,105,126,

150-153,180,182-186,210-212,214, 231,280

Adams, R. A. 7,14,155,231 adjoint operator 47,319 Ahlfors mollifier 280, 324 Ahlfors, L. V. 49,280,327 Aikawa, H. 150,153 AIssaoui, N. 49 (o:, p )-capacities - relations between 129,148,151,153 (o:, p)-capacity VI, 19,20,29,44, 153

see a/sa capacity - and coverings 145 - and Hausdorff content 152 - and Hausdorffmeasure 129,151 - history 49 - lower estimate in terms of Hausdorff

content 137,139 - of baII 131 - of Cantor set 143, 144 - of manifold 139 - quasiadditivity of 150 - sharpness of comparison 146, 147 - under contraction 152 - under Lipschitz mapping 140,141 - upper estimate in terms of Hausdorff

content 134 (o:, p )-fine 176 see alsa fine (o:, p)-fine topologies - comparison of 184 (o:, p )-fine topology 186 see also fine

topology (a, p)-quasicontinuity 156 see a/sa

quasicontinuity (o:, p )-quasicontinuous representative

156,158,159 see also quasicontinuous representative

(o:, p)-quasieverywhere, (o:, p)-q.e. 20 see alsa quasieverywhere

(o:, p )-thick set see thick set (o:, p )-thin set see thin set analytic q -capacity 311 analytic capacity 311,324,326 analytic set see Suslin set Ancona, A. 82 Anger, B. 211 annihilator 306 - of Dq(E) 313 - of L~(G) 306,308,326 - of L~(G) 319 - of Rq (E) 309 - of W;,·P (KC) 235 approximation - by analytic functions -- in U 305,316 - - in U, and (1, p )-stability 310 - by harmonic functions 279 -- in U 305,318 -- in U and (2, p)-stability 322 -- in WI.q 316 -- in WI.q and (1, p)-stability 312,313 - - uniform 326 - by potentials in U 309 - bypotentialsin W l .q 314 - by rational functions 305 - - constructive 325 -- in LI 324 -- in Lq 309 -- uniform 311,324--326 - by solutions of elliptic equations 318 - - and bounded point evaluations 325 -- in U and (0:, p)-stability 322 - local degree of 85, 122 - of C"'-functions vanishing on a set 233 - of functions with vanishing trace 233,

234,242,281,302 - of nonnegative functions in W~n.p (Q)

235

352 Index

- one-sided see one-sided approximation - Sobolev problem 278 area of (N - I)-dimensional unit sphere

8 Aronszajn, N. 10-13,49,152,153,185 Arzela-Ascoli theorem 197 atom VII, 123, 127 - Cs_ 111 - smooth 85,126,281,282 atomic - nonlinear potential 119-121,186 -- and Wolffpotential 119,121 - nonlinear potential theory 127 - potential 117 - representation 85,111,113,187,281,

282,296 - - and quasicontinuity 283

Babuska, 1. 327 Bagby, T. 8,185,211,279,303,325-327 balayage 184,308 Baras, P. 51 Bauman, P. 182 Beckner, W. 82 Benkirane, A. 49,82 Berger, M. 214 Bers, L. 280, 324, 326 Besicovitch, A. 153 Besov capacities - relations between 152 Besov capacity 105, 167 - and Hausdorff content 152 - dual definition 106,107,118 Besov nonlinear potential 106, 107, 126,

167, 168 Besov space VI, 80, 85, 86, 125, 155,302 - atomic representation 111, 113 - equivalent norm 87,89, 111, 113 - homogeneous 86 - potential representation 105 - representation theorem 90,104 - weighted 97 Besov, O. V. 125 Bessel capacity 38, 131 - compared to Riesz capacity 131, 150 Bessel function Il Bessel kemel 9,50,193,198 - analyticity of Il - as elementary function 12 - asymptotic formulas 11,12 - compared to Riesz kemel 10 - derivative of 13 - group property 10, 21 - integral representation 10

- Laplace transform representation 12 - positivity 10 - representation by Bessel function Il Bessel potential 10 - derivatives 57 - integral estimate 56 - name of II - nonlinear 38 - representation as 20 Bessel potential space VI, 13 - as Lizorkin-Triebel space 92,204, 281 - dualof 13 - equivalent norm on 122 Beuriing, A. V, 49, 50, 81, 82,153,279,

327 BMO, functions of bounded mean

oscillation 79,92, 126 Boccardo, L. 82 Bochner, S. Il Borel property of e",p(E) 174 Borel sets, capacitability of 28 Borichev, A, A, 150 boundary regularity 165,166,181-183 - modulus of continuity 182 boundary values 129,142,153,161,181 bounded mean oscillation 79,92,126 bounded point evaluation 325 bounded variation 181 boundedness principle 39,40,50, 190,

214,264 Bourdaud, G. 80 Branson, Th. P. 81 Brelot, M. 150, 152, 185, 186, 278 Brennan, J, E. 326 Brezis, H. 67,82,238,280 Browder, F. E. 67,82,238,280 Brownian motion 48 Burenkov, V. 1. 279 BV, functions of bounded variation 181

Caffarelli, L. 50 Calderon extension theorem 7, 142,238 Calderon theorem 13,20 Calderon, A. P. 6,7,10, 13,238 Calderon, C. P. 185 Calderon-Zygmund theorem 6,94 Campanato space 79,151 Cantor set 129, 143, 147, 149 - (a, p)-capacity of 143,144 - Hausdorffmeasure of 143,153 capacitability 49 - for Hausdorff content 138 - of Borel sets 28 - of Suslin sets 28

~ theorem 28 capacitable set 28 capacitary function 21,28,35~37,

106-108,119 capacitary integral 212, 230 ~ sublinearity of 211 capacitary measure 21,35,37,108,151,

165,168,170,171,229,264 ~ outer 37,168 capacitary potential 21,28, 174,264,312 ~ boundedness of 41 capacitary strong type inequality 181,

187,189,199,200,209 ~ failure of 209 ~ history of 213 ~ in Lizorkin~ Triebel space 204 capacitary weak type inequality 188 capacity V, 17, 18 ~ (a,p)- VI,19,20,29,44,153seealso

(a, p)-capacity ~ LP-, for general kemel 25,104 ~ N",p 45 ~ analytic 311, 324, 326 ~ analytic q- 311 ~ and algebraic properties of set 226 ~ and Lebesgue measure 38,39 ~ and stochastic processes 48,51 ~ atomic 118 ~ Besov 104,105,167 ~ Bessel 38, 131 ~ condenser 209,312 ~ distribution 45 ~ dual definition 21,34,35,106,107,

118,220,311,326 ~ equivalence of C",p and C( , ; FP,q)

107 " ~ equivalence of C",p and C( , ; f.p,q)

118 " ~ equivalent 20,65,105 ~ for baU 129 ~ in spaces with mixed norm 107 ~ instabilityof 314,324,325 ~ Lizorkin~ Triebel 104, 105 ~ ~ dual definition 107, 118 ~~ equivalence with (a, p)-capacity 203 ~ maximizing property of 36 ~ Newton 17 ~ outer 19,25,153 ~ parabolic 214 ~ polynomial 226,322,327 ~ probabilistic interpretation of 48, 51 ~ relative 223,241 ~ Riesz 38,131 ~ scale invariance of 167, 224

Index 353

~ sets of zero 26,263,266,271 ~ strongly subadditive 211,212 ~ subadditivity of 26 ~ triviality of 38, 189 CarIeman, T. 325 Carleson, L. VI,40,,48~51,81, 138,

151~153, 186,326,327 CarIsson, Anders 180, 231, 301 Cartan, H, V, 326 Cauchy, A, II Cauchy~Riemann equation 306 Cauchy~Riemann operator 47,306 chain rule 62, 239 Chandrasekharan, K. 89,91 Chang, S,-Y, A, 81,82 characterization of f E L~,{J(E) 281,302 characterization of f E W~"'P (Q) 234 Choquet property 1:55,165,175,185 Choquet theorem 28, 138 Choquet, G, 28,49, 185,211,327 Cioranescu, D, 325 Clarkson inequalities 14 Clarkson theorem 14 Clarkson, 1. A. 14 cIassical potential theory V, VI, 17, 18,

20,21,39,41,42,48,50, 151~153, 155, 164,184-186,308

cIosure of QE, description of 27 Cohn, W, S, 126 Coifman, R, 83 coincidence of positive cones 103, 126 compact imbedding 227 compactness ofimbedding 187,195,198 comparison theorem 134,137,139 complex interpolation 79 condenser capacity 209,312 conformal invariance 49 connectedness ~ arcwise 184 ~ in fine (quasi-) topology 184 ~ local 184 continuity ~ of imbedding 187,191,193,210 ~ of restriction to subspace 142 continuity principle 41, 50 contraction 62, 82, 152 convex set, weak cIosure of 15 convolution 4 ~ maximal function estimate of 102 covering lemma ~ Maz/ya~Preobrazhenskii 211 ~ Netrusov 302 ~ simple Vitali 16

354 Index

- Whitney 16 see a/so Whitney decomposition

Dahlberg, B. E. 1. 80, 82, 83, 2 \3 DaI Maso, G. 182,325 Daubechies, 1. 127 Davie, A. M. 184 Davis, Burgess 184 degree of approximation, local 85,122 Dellacherie, C. 49 Deny, J. V, 7,10,50,82,153,185,186,

278,279,307,308,326 derivative - ofproduct 158,241 - of potential 57,259 Diestel, 1. 15 differentiability 164,303 - fine 184 differential - and zero trace 287 - of order s 282,283,293,299,323 Dirac distribution, measure 8, 228 - sweeping of 308 Dirichlet integral 327 Dirichlet principle V Dirichlet problem V, VIII, 164, 165,

181-183 - and (1, 2)-stability 312 - fine 278 - for p-Laplace equation 166 - for polyharmonic equation 183, 236,

237,278 - in domains with many holes 325 Dirichlet space 82, 279 distinguished element 155 distinguished representative 187 distribution V,4 - capacity 45 - in W-m,p', nullspace of 235 - in W-m .p', approximation of 235 - of order m, nullspace of 233 - positive 4, 22 - temperate (tempered) 4 divergence of positive series 288, 303 Donchev, T. 183 Doob, J. L. VII, 48, 185 Du Plessis, N. 48 dual approximation problem 306 dual definition of N".p(K) 45 dual definition of capacity 21,34,35,

106,107,118,220,311,326 dual space - of LI(A~,,) 212

- of Besov space B:·q 87 - of C(K) 2 - of Co(.Q) 2 - of LP(C".p) 202 - of LP(E) 2 - of BesseI potential space L a.p 13 duality between L-"'P' and L"'P 188 dyadic - ball 1 - cubes 16,73,76,111,246,282,302 - Hausdorff measure 302 - sequences 116 Dynkin, E. B. 48,51 Dyn'kin, E. M. 303

Egorov theorem, extended 26 elliptic equation 182 - degenerate 182,185 - in non-divergence form 182 - of higher order 183 - quasilinear 48,51,166,182 - semi-linear 183 elliptic linear operator 47,65,319 embedding see imbedding energy integral 17, 120 - generalized 22,36,106-108,119,120 energy, mutual 24, 105, 117, 119 equilibrium measure 17 equilibrium potential 17 equivalence - of C",p and C(·; F:,q) 107 - of C",p and ce·; t:,q) 118 - of C".P and N".p 65 equivalent capacities 20,65, 105, 153 equivalent norm - on L'".2 68 - on L"'P 72, 122 - on Besov space 87,89,111,113 - on Lizorkin-Triebel space 97 102

111,113 ' ,

equivalent sets 176 Evans, G. C. 50,152,153,185 Evans, L. C. 152,181,182,238 exceptional set 49 exponential estimate of potential 55,56,

81,210 - sharp 58,81 extension - of Lipschitz mapping 140 - domain 7,227 - operator 7,227 exterior cone condition 245 extremal - function 18,20,21,46,106-108,167

- - in Moser inequality 81 - length 49 - measure 35, 108, 119 - problem 48 - sequence 119

F",p-thin set 183 Fabes, E. B. 182, 185 Falconer, K. 1. 152 Fan K. 50 Faraday cage 327 Federer, H, 140,181,185,325 Fefferman, C. 3,4,83,210 Fefferman, R. 212 Fefferman-Stein theorem 3,98,101,102,

104,112,116,122,125,207,282,285 Femstrom, C. 324, 325 fine - boundary 184 - closure 176,184 - - capacity of 179 - continuity 155,156,176,185,186,318 -- and quasicontinuity 177,178 -- explicit condition for 180 - differentiability 184 - differential 184 - Dirichlet problem 278 - interior 176 - neighborhood 164,176 - topology 156, 164, 176 - - comparison of 184 - - connectedness in 184 finely - closed set 176 - isolated point 176 - open set 176 Folland, G, B. 7,20,236,279 Fontana, L. 82 Forsling, G. 302 Fourier inversion formula 5 Fourier transform 5 Fourier transformation 5 fractional maximal function 3,54,210 - and capacity 151 - and potential 72,75,78 Frazier, M. 80,126,127,212,214 Friedrichs, K. 278 Frostman theorem 129,136,151,193 Frostman, O. V, 40, 48, 136, 152, 156,

185,186 Fuglede, B, VI,24,49,50, 127, 152, 153,

184-186,278,279 Fukushima, M. 48 functions that operate 62

Index 355

- on g"L~ 68 - on g"L~ 63 - on I"L~ 65 - on L"'P 80 - on nonlinear potential 262 - on w",p 62 - on W~,p 80 fundamental solution 8,9,47,319

Gagliardo, E. 79 Gamelin, T. W. 326 Garcia-Cuerva, 1. 83 Gariepy, R. F. 152,181-183,238 Garofalo, N. 182 Garroni, A. 325 Garsia, A. M. 81 Gauss theorem 17 Gauss, C. F. V, 17,48 Giaquinta, M. 79,80 Gilbarg, D. 81,82 Gol'dshtein, V. M. 152 Gonchar, A. A. 324,327 good A inequality 72, 73 Gossez, J.-P. 82 Green function 308,312 Grun-Rehomme, M. 208

Hadamard, 1. 303 Hahn decomposition theorem 208 Hahn-Banach theorem 14 Hanin, L. G. 303 Hanner, O. 15 Hansson, K. 214 Hardy space VIII, 92,126 Hardy, G. H. 3,81 Hardy-Littlewood maximal function 3 Hardy-Littlewood-Wiener theorem 3 Hardy-Sobolev space 80, 303 harmonic measure VIII, 312 Hamack property 257,259,262 Harvey, R. 50,51 Hatano, K. 153,184 Hausdorff - capacity 132 -- dyadic 212 - content 132, 136, 325 -- and (a, p)-capacity 134,137,139,

152 -- and Besov capacity 152 - - and quasicontinuity 180 - dimension 133 - measure VII, 132, 152,237,301 -- and (a, p)-capacity 129,151 - - dyadic 302

356 Index

-- of Cantor set 143,153 Havin, V. P. VI, 49-51, 152, 153, 185,

279,324,326,327 Heard, A. 183 heat equation 182 Hedberg, L. 1. 50,67,79,81,82,126,

127,153,183-186,231,278-280,302, 325-327

Heinonen, 1. VIII, 48, 184, 185 Heisenberg group 182 Helms, L. L. 48,185 Hempel, 1. A. 81 Hestenes theorem 7, 142 Hestenes, M. R. 7 Hewitt, E. 14 Hilbert transform 5 Hilbert, D. V Hirschman, I. 1. 79 HOider space 79 homogeneous seminorm 8 Hormander condition 6, 94 Hormander, L. 2,4--6,8,47,50,93,125,

126,278,307,319,326 hypoellipticity 319

ideal 293, 294 - in C 293,303 - in C III 281,293,303 - in Sobolev algebras 303 - local 293 - Whitney theorem 281,293,303 imbedding - compactness of 187,195,198 - continuity of 187,191,193,210 - of W~·P(G) in LI"" 307 - of W,,·l 212 - of Morrey-Campanato space 79 - upper triangle case 211 Indiana University Union 82 inhomogeneous maximal function 3, 210 instability of capacity 314, 324, 325 integrability with respect to capacity 187,

212 integral inequalities 56 interior cone condition 243 interpolation inequality 56,57,78 irregular set 164

Janson, S. 80 Jawerth, B. 83,126,127,212,214 Jerison, D. S. 182 John, F. 8,20,47 Jones, P. W. 7,280 Jonsson, A. 214

Kothe, G. 14 Kahane,1.-P. 51,82 Kaneko, H. 48 Kateb, M. E. D. 80 Katznelson, Y. 89,91,279 Keldysh, M. V. 312,326 Kellogg property 155,165,175,185,238,

241,277,280,282,318,324,327 Kellogg, O. D. 48,185 Kenig, C. E. 182 Kerman, R. 83,153,210,214 kemel 24 - Bessel see Bessel kemel - general 24 - Newton 8 - on RN x RN X N 104 -onRNxNxZN 117,121 - radially decreasing convolution 38,

129,134,135,187,189,191,195,214 - regular 50 - Riesz see Riesz keme1 - singular Riesz 5 - truncated 196 Khanin, L. G. see Hanin, L. G. Khavin, V. P. see Havin, V. P. Khinchin, A. Ya. 93 Khinchin inequality 93,96,125 Khruslov, E. Ya. 325 Khvoles, A. A. 326 Ki1pe1ainen, T. VIII, 48,166,182,183,

185 Kinderlehrer, D. 50,82,279 Kirszbraun, M. D. 140 Kneser, H. 50 Kolsrud, T. 278,325 Kuznetsov, S. E. 51

de La Vallee Poussin, Ch.-1. V Lanconelli, E. 182 Landes, R. 82 Landis, E. M. 182 Landkof, N. S. 9,40,48,49, 150, 185,

211 Laplace equation V, 165 - p- VIII, 166, 182 Laplace operator 2 - p- VIII, 20 - powers of 5,9,47 Laplacian see Laplace operator Laurent expansion 311 Lavrent'ev, M. A. 312 Lebesgue point 158,180,181,185,219,

234,239,246,283,299,301,315 - and atomic representation 282

- in U -sense 159 Lebesgue, H. 50 Leibniz mie 158,241 Levi, Beppo 7 Lewis, John L. 126,184 Lindberg, Per 325,327 LindelOf property, quasi- 184 LindelOf, E. 325 Lindqvist, P. 182, 183 Lions,1. L. V, 7,50,82,185,186,278,

302,307,308,326 Lipschitz domain 7 Lipschitz mapping 129,152 - extension of 140 Littlewood,1. E. 3,81 Littlewood-Paley theorem 92 Littman, W. 50,51,82,181,182 Lizorkin, P. I. 125 Lizorkin-Triebel capacity see capacity,

Lizorkin-Triebel Lizorkin-Triebel nonlinear potential 108 Lizorkin-Triebel space VII, 80, 85, 91,

125,233,301-303 - atomic representation 111, 113 - Bessel potential space as 92, 281 - equivalent norm 97, 102, III, 113 - homogeneous 92 - potential representation 105 - representation theorem 103, 104 - strong type inequality in 204 - trace space of 212 local degree of approximation 85, 122 Loewner, C. 49 Loomis, L. H. 303 lower semicontinuity 3 Lukes, J. 186 Luzin, N. N. 50 Lyons, T. 184 Lysenko, Yu. A. 324,327

Macdonald function II Magenes, E. 302 Maly, J. 166,181-183,185,186 Malgrange, B. 303 Malliavin, P. 48,279 Marchenko, V. A. 325 Marcinkiewicz multiplier theorem 126 Marcinkiewicz, J. 126 Maria, A. 1. 40 Marschall, 1. 302 Marshall, D. E. 81 Martio, O. VIII, 48, 182, 183, 185 Mateu, 1. 302,303,325 Mattila, P. 325

Index 357

maximal function 3,54, 159,260,284 - estimate of convolution 102 - fractional see fral;tional maximal

function - Hardy-Littlewood 3 - inhomogeneous 3,74,210 - Peetre 97 - vector valued 3 maximum principle 40, 82, 279 Mazur lemma 15,27,266,274,276 Maz'ya, V. G. VI, VIII, 7,14,48-51,

79,80,82,152,153,166,181-183,185, 209-215,226,231,303,326,327

Maz'ya-Kilpeliiinen-Maly theorem 166, 182

measure - absolutely continuous w. r. t. (a, p)-

capacity 187,208 - capacitary see capacitary measure - equilibrium 17 - in (B~~q)+ 90, 110, 126 - in (F~~q)+ 102,103, 110,126 - in (L -a. p )+ 103,110,126 Mel'nikov, M. S. 326 Mergelyan, S. N. 326 Meyer, P.-A. 49 Meyer, Y. 80,127 Meyers, N. G. VI, 7, 49, 50, 79,127,152,

153,183,185,186,231,280,323 Michael, J. H. 181 Mikhlin, S. G. 85,93, 126 Mikhlin-Hi:irmander multiplier theorem

93, 125, 126 minimax theorem 30,34,49,50 minimizing sequence, convergence of 15 Minkowski inequality 4 Minty, G. J. 140 mixed norm 3,78,86, 104, 117 Mizuta, Y. 184 modified Bessel function of the third kind

Il module, closed Ci)- 293-295 modulus of continuity 182 modulus of curve family 49 Molchanov, A. M. 231 monotone norm 202 Morawetz, C. 15 Morrey, C. B., Jr. 7 Morrey-Campanato space 79, 151, 214 Morris, G. R. 81 Mosco, U. 182 Moser inequality 81 Moser,1. 81 Muckenhoupt Aoo condition 83

358 Index

Muckenhoupt, B. 53, 83 Muckenhoupt-Wheeden theorem 72,83,

103,107,126,214 Mulla, F. 12,13,49 multi index 2 - length of 2 multiplier 66,67,241,281,282 - construction of 242,243,247,291 - theorem - - of Marcinkiewicz 126 -- of Mikhlin-Hi:irmander 93,125,126 multiresolution analysis 127 Murat, F. 82,325

Nagel, A. 181 Netrusov, Yu. V. VII, VIII, 126, 127, 150,

152,153,164,181,209,211-214,233, 281,294,301-303,327

von Neumann minimax theorem see minimax theorem

von Neumann, J. 50 Newton capacity 17 Newton kemel 8 Newton potential 8, 17 Nieminen, E. 211 Nikol'skii, S. M. 88,125 Nilsson, Per 126 Nirenberg, L. 79 non-increasing rearrangement 60 nonlinear potential VI, 21, 36, 46, 49, 166,

167,169,175,183,186 - atomic 119-121,186 - Besov 106,107,126,167,168 - Bessel 38 - estimate of derivatives 259 - Hamack property 257,259,262 - Lizorkin-Triebel 108 - lower estimate 44,45,257,258 - Riesz 38 - smooth truncation of 262, 264, 272 nonlinear potential theory VI, VIII, 18,

48,49,85,233 - atomic 85, 127 nontangential limit 161 nonvanishing condition 89,102 norm - monotone 202 - on space of quasicontinuous functions

199 Nystrom, K. 226, 327

O'Neil lemma 60 O'Neil, R. 61,81 obstac1e problem 50, 182

Ohtsuka, M. 153 0ksendal, B. 184 Olivier, L. 303 one-sided approximation 66, 82, 238 - in Orlicz-Sobolev spaces 82 - parabolic 82 operator - va 68,83 - - pointwise estimate of 69 - fa 122 - sa 72,83, 122 Orlicz-Sobolev spaces 82 Orobitg, J. 302,303,325 outer capacity 19,25,153

p-Laplace operator VIII,20 p-Laplace equation VIII, 166, 182 p-superharmonic function 182 pairing 4 parabolic equation 182 parabolic potential 214 Parseval formula 5 partial differential equation - boundary regularity 165,166,181-183

see a/sa Wiener criterion - elliptic see elliptic equation - linear second order 20 - nonlinear VI, 20, 67, 82 - parabolic 182 - quasilinear VIII, 48,51,166,182,183 - removable singularity see removable

set partition of unity 86 Peetre maximal function 97 Peetre theorem 87,97 Peetre, J. 85,87,89,97,125,126 Perez, C. 83, 126 Perron method 48 Phong, D. H. 210 Pierre, M. 51 Pisarevskii, B. M. 324, 327 Plancherel theorem 5 Poincare inequality 8,215,231 Poincare type inequality 215,238 - and c10sed graph theorem 226 - and thickness 245,267 - depending on capacity 225,226,241 - - best constant in 231 point of absolute convergence 283,287 pointwise bounded approximation 66,

238 pointwise estimate of potential 54,55 Poisson integral 161, 181 Poisson, S. II

polar set 50 Polishchuk, E. M. 303 Polking, J. C. 50,51,82,83,279,325,

327 polynomial approximation - local 122 - of functions vanishing on a set 223,

229 polynomia] capacity 226, 322, 327 Pompeiu formula 307 porous set 212 positive cone - coincidence of 103, 126 - in B~~q 90,110, 126 - in F!~q 102,103, IlO, 126 - in U(E) 2 - in L -a,p 103, IlO, 126 - in M(.Q) 2 - of distributions 4 positive distribution 4 potential 24 - and fractional maximal function 72,

75, 78 - atomic 117 - average of 43 - Bessel see Bessel potential - capacitary see capacitary potential - compared to surn 77, 78 - continuityof 41,42 - equilibriurn 17 - Newton 8, 17 - nonlinear see nonlinear potential - parabolic (heat) 214 - Riesz see Riesz potential - Wolff see Wolff potential PreobrazhenskiI, S. P. 211 principal value integral 5 product rule 158,241 projection V, 227, 228

quasi-LindelOf property 184 quasiadditivity of (a, p)-capacity 150 quasiclosed 178 quasiconformal mapping 49 quasicontinuity 155,156,179,186,317 - and atomic representation 283 - and fine continuity 177, 178 - and Hausdorff content 180, 181 - Holder type 181 - of Besov potential 168 quasicontinuous function - integrability of 187,188,212,227 - measurability of 187,188 quasicontinuous functions, space of 200

Index 359

quasicontinuous representative 156, 158, 159,180,181,185,219,234,281

- integrability of 187,188,212,227 - uniqueness of 157,158,178,185 quasieverywhere 20 - (g, p)- 26 quasilinear equation VIII, 48, 51, 166,

182,183 quasiopen 178, 317 quasitopology 178 - connectedness in 184

Rademacher functions 92,95 Radon measure 2 regular point 164,181 - Wiener criterion for 165, 166 see also

Wiener criterion relative capaeity 167,223,241 removable set 47,51,65,306,326,327 - for Wm,p 237,278 representation formula 308 - for functions vanishing on a set 219,

222 - Sobolev 217 Reshetnyak, Yu. G. VI, 49, 152, 153 Rickman, S. 182 Riemann theorem 51 Riemannian manifold 82 Riesz capacity 38, 131 see also capacity Riesz kemel 9,193,198 - integral representation 10 - modified 74,223 - singular 5 Riesz potential 9 - derivatives of 57 - integral estimate of 56 - nonlinear 38 - pointwise estimate of 54,55 - weak type estimate of 56 Riesz potential space 13 Riesz transform 5 Riesz, M. V,5 Riviere, N. M. 185 Rogers, C. A. 138,152 Rubio de Franeia, J. L. 83 Rudin, W. VI, 2, 5,14-16,25,140,181,

197,208,303,305,307,312 Runge theorem 305

Saak, E. M. 327 Sawyer, E. T. 83, 153,210,214 Schechter, M. 214 Schrodinger operator 213 Schulze, B.-W. 279,327

360 Index

Schwartz class 4, 85 Schwartz distribution 4 Schwartz, L. V, 2, 4,5,12,50,158,279,

306,319 Segala, F. 182 segment property 278 semicontinuity 3,24,121 seminorm, nuH space of 227 sequence space 116, 117 Serrin, J. VI, 7,48,49,51,153 set - analytic see Suslin set - Borel see Borel set - of capacity zero 26,263,266, 271 - of infinities of potential 26,49, 153 - ofnon-uniqueness for La,p 314 - ofuniquenessforLa,p 314 - porous 212 - stable see stable set - Suslin see Suslin set - thick see thick set - thin see thin set Shapiro, Harold S, 125 Shapiro, Joel H. 181 Shaposhnikova, T. O. 212,213,303 sharp exponential estimate 58, 81, 82 sharp inequality 81 simple Vitali lemma 16 Sinanyan, S. O. 326 singular integral operator 6 singular Riesz kemel 5 singularity, removable see removable set Sion, M. 49 Sjodin, T. 49,126,185 Skrypnik, I. V. 183 Smith, K. T. 10-13,49,152,153,185 smooth atom see atom smooth truncation 63,80,150,209,213,

240 - of non-negative functions 80 Sobolev algebras 303 Sobolev imbedding theorem 14,56,80,

155 Sobolev inequality 81, 218 Sobolev representation formula 215,217 Sobolev space V-VIII, 6, 233 - homogeneous version 8 - pointwise representation 8,215,217 - potential representation 9,10,13,20 - weighted 185 Sobolev trace theorem 236,278 Sobolev uniqueness theorem 237,278 Sobolev, S. L. 7,14,81,155,231,236,

237,278

space of quasicontinuous functions 200 - minimality of 202 spectral measure 50 spectral synthesis 279 - 2m- 279 - counterexamples 279 spectrum 279 spherical h-density 151 stability 305 - and approximation by harmonic functions

312 - and approximation in L~ 310 - and Dirichlet problem 312 - and thickness 320 - definition of (Il, p)- 319 - necessary and sufficient condition for

(Il, p)- 320,322 - necessary condition for (Il, p)- 320 - of nowhere dense set 314 - sufficient condition for (Il, p)- 320 stable set 309,313,314,316,319,320,322 - example of non- 314,321 Stampacchia, G. 79, 82, 181, 182, 279 Stein, E. M. VI, 3-11,13,14,16,56,83,

92,125,126,159,161,185,189,192, 238,263,265,315

stochastic processes 48, 51 Strichartz theorem 72,85,122,127 Strichartz, R. S. 81,83,127,208 Stromberg, K. 14 strong subadditivity 211, 212 strong type inequality see capacitary

strong type inequality subadditivity of capacity 26 - strong 211,212 submodule of La,p 293,294 Suslin set 35,49, 170 - capacitability of 28, 29 Suslin, M. Va. 50 sweeping 184, 308 Szeptycki, P. 12, 13,49

Tarkhanov, N. N. 51,325,326 Tauberian condition 89, 102 Taylor formula 215,220 Taylor polynomial 2, 228, 282,283 - as ring homomorphism 293 Taylor, S. 1. 151-153 temperate (tempered) distribution 4 test function 4 thick set 164,166,241,266,320 - uniformly 242,243 thickness - and Poincare type inequality 245

- and zero trace 246 thin set 155,164--166,182,183,186,241,

316 - Brelot characterization of 155,165,

168, 185, 186 - definition of (a, p)- 166 - Fa,p- 183 - history of 185 - necessary and sufficient condition for

168,225 - Wiener criterion for 165,166 thinness, weighted 185 Thomson, J, E. 326 Tietze extension theorem 187,202 Titchmarsh, E, C. Il topological degree theory 82 Torchinsky, A. 4 trace 125,155,219 - of function in [U'P 158,234 - of potential 191-193,195,198,210 - - exponential estimate of 210 - on boundary 236 trace inequality 187,191,193,210,214 - history of 213 - upper triangle case 211 trace operator 234,287,301,318 - kemel of VII, 233, 234, 281 trace space, invariance of 212,213 trace theorem, Sobolev 236, 278 Treves, F, 279 Triebel-Lizorkin space see Lizorkin-

Triebel space Triebel, H. 125,126,302 triviality of capacity 38,189 triviality of LZ(G) 306,326 Trudinger inequality 81 Trudinger, N. S. 81,82 truncated kemel 196 truncation 62,80,82,213,239,279,317,

324 Turesson, B. O. 83,185 two weight inequalities 83

Ugaheri, T. 50 uniform boundedness principle 14 uniform convexity 14,22,46 uniformly thick set 242, 243 uniqueness theorem - for fine Dirichlet problem 278 - for higher order Dirichlet problem 236,

237,278 - of Brelot 278 - of Sobolev 237,278 uniqueness, set of 314

Index 361

unspecified positive constants

Viiisiilii, 1. 49 value in sense of Lebesgue 283 see also

Lebesgue point Vasilesco, F, 50 Verbitsky, 1. E. 83, 126,210,211 Verdera, J. 302,303,326 Vitali lemma, simple 16 Vitushkin approximation scheme 325 Vitushkin, A. G. 324,326,327 Vivaldi, M. A. 182 Vodop'yanov, S. K. 49,185 von Neumann minimax theorem see

minimax theorem von Neumann, 1. 50

Wallin, H. 151,153,185,214 Walsh, J. L. 325 Watson, G, N. 11,12 wavelet 127 weak closure of convex set 15 weak derivative 7 weak type capacitary estimate 159,169,

264,272 weak type inequality 56,188,189,192 weak* compactness 14 weak* L OO 279 weak* topology 2 Webb, 1. R. L. 82 weight function 42,279,280,324 weighted inequalities 210 weighted Sobolev space 185 weighted thinness 185 Weinberger, H. 181,182 Weiss, G. 4, 127 Welland, G. 83, 126 Wermer, 1. 48 Weyl lemma 306, 313, 319 Wheeden, R. L. 53, 83 Whitney decomposition 16,73,75,263,

265,272 Whitney ideal theorem 281,293, 303 Whitney, H. 16,281,293,303 Wiener criterion 165,185,186,282 - extensions of 181 - for higher order equations 183 - for nonlinear potentials 183,185,186 - for obstacle problems 182 - for parabolic equations 182 - for quasilinear equations 166,182,183 - of Maz'ya-Kilpeliiinen-Maly 166,182 Wiener lemma 16 Wiener Tauberian theorem 89

362 Index

Wiener theorem 89,91,125 Wiener, N. 3,16,165,185 Wildenhain, G. 279,327 Wolff inequality 85,109,126,138,140,

153,186,195,198,214 Wolff potential 45,109,110,167,168,

323 - and atomic nonlinear potential 119,

121 - and fine continuity 180 - and zero trace 246 - homogeneous 109, 110

- local version 183 Wolff, Th. H. 85,126,127,175,186,280,

327

Yang, P. C. 81,82 Young inequality, W. H. 4 Yudovich, V. 1. 81, 210

Ziemer, W. P. VIII, 7, 14,61,152,164, 181-183,185,215,226,231,238,303

Zygmund, A. 4,6,81,126

List of Symbols

Symbols listed in order of appearance

Chapter 1

A various unspecified constants, 1

N natural numbers, 1

Z integers, 1

R real numbers, I

RN Euclidean N -space, 1

x . y = x,y, + ... + XNYN Euclidean scalar product, I

Ixl = (x . x)'/2 Euclidean norm, 1

E c10sure of E, 1

E", RN \ E complement of E, I

EO, int E interior of E, 1

IEI, dm, dx Lebesgue measure, l

XE, X (E) characteristic function, 1

B(xo, r) open baII, 1

BII(xo), Eli dyadic baII, 1 !+,!_ positive and negative parts of !, 1

(a/aXj)! = aJf partial derivative, 2

V f = (ad, ... , aN j) gradient,2

.1f = (a~ + ... + a~)f Laplacian,2

a = (a" ... ,aN) multiindex, 2

la I = a, + ... + UN length of multi-index, 2

D" f partial derivative, 2

x" = X~' .. . x"J 2

a! = al!" ·aN! 2 P;"f (x + y) Taylor polynomial, 2

VIII f higher gradient, 2

I Vlllf I 2

U (E), U, U (v) Lebesgue spaces, 2

IIfIILP(EJ, IIfll p , U-norms 2 L~(E) positive cone in U(E), 2

u' (E), p' = p/(p - 1) dual space of U(E),2

C(E) space of continuous functions, 2

Co(Q) compactly supported functions, 2

Cm (Q) space of m times differentiable functions, 2

Cg' (Q) compactly supported functions in cm(.Q),2

M(Q) dual of Co(!J), Radon meas­ures,2

M+(Q) positive measures, 2

Mf, Mof Hardy-Littlewood maximal function,3

Ma f fractional maximal function, 3

Ma,of inhomogeneous maximal function, 3

U(lq) 3

f * g convolution, 4 COO(Q) 4

Cg"(Q) compactly supported test functions, 4

S Schwartz c1ass of test functions, 4

V' space of distributions, 4

S' space of temperate (tempered) distributions, 4

( " .) pairing, 4

:F Fourier transformation, 5

:F f, Î Fourier transform, 5

Rj Riesz transform, 5

Rj singular Riesz kemel, 5 Wa,P(Q), Wa,p Sobolev space, 6,7

W;,p (.Q) Sobolev space, 7

w;'P (il) homogeneous Sobolev space, 8

364 List of Symbols

WN-l area of (N - l)-dimensional unit sphere,8

h Newton kemel, 8

Ia Riesz potential operator, 9

la Riesz kemel, 9

Ya constant in la, 9 G a Bessel kemel, 9

(Ia Bessel potential operator, 10

Jv Bessel function, Il

K v modified Bessel function, Macdonald function, 11

La,p Bessel potential space, 13

ia,p Riesz potential space, 13

p* Sobolev exponent, 14

Chapter 2

C(,) capacity, 17

C~,p, a E N capacity, 19 Llp p-Laplace operator, p-Laplacian, 20

WK subset of S, 20 Ca,p, a > O, (a, p)-capacity, Bessel

capacity, 20, 38

Cih c10sure of WK, 20 F K capacitary potential, 21, 22

fK capacitary function, 21, 22

IL K capacitary measure, 21

V:' p nonlinear potential, 21

M measure space, 24

(1 f, 9 IL potentials, 24

&g mutual energy, 24

[h subset of U(v), 25 Cg,p capacity, 25

il E c10sure of ilE , 27 V:,p nonlinear potential, 36

Ca,p, a > O Riesz capacity, 38

V:'p Riesz nonlinear potential, 38

W:' p Wolff potential, 45

W:' p Wolff potential, 45 WK subset of S, 45 Na,p distribution capacity, 45

C partial differential operator, 47

[,* adjoint operator, 47

Chapter 3

p* Sobolev exponent, 56 f* non-increasing rearrangement, 60 f** average of f*, 60 [a] integral part of a, 68 va operator, 68 sa operator, 72

la,8 modified Riesz kemel, 74 U([q) 78

Cp, Morrey-Campanato space, 79

BMO functions of bounded mean oscil1ation, 79

Chapter 4

Bg,q Besov space, 86 lq (U) 86, 104, 117

I,JJ space of polynomials, 86 Bg,q homogeneous Besov space, 86 F :,q Lizorkin-Triebel space, 92 p:,q homogeneous Lizorkin-Triebel

space,92 {rn(t)}O' Rademacher functions, 92

B:;'f weighted Besov space, 97 u" Peetre maximal function, 98

(l1:*U}O' Peetre maximal function, 98 U(lq) 104,117

ha (x. y, n) kemel, 104

'Ha f, Ha IL potentials, 104 C(·; Bg·q) Besov capacity, 105 C(.; F:,q) Lizorkin-Triebel capacity,

105 &ha mutual energy, 105

V:' p Besov nonlinear potential, 106

V:, p,q Besov or Lizorkin-Triebel nonlinear potential, 107, 108

W:' p Wolffpotential, 109, 110 W:' p homogeneous Wolff potential, 109,

110

Qnk dyadic cubes, III Xllk characteristic function for Qllk. 111 l(Q) side of cube, 111, 246 A Q concentric cube, 111

Qnk = 3Qnk expanded cube, 111

Xnk characteristic function for Qllk. III allk atoms, 111

sn(x) = LkEZN SnkXllk(X) III ~a(x,n,k) kemel,117

SJas, iJaţt atomic potentials, 117

ţtllk = ţt(Qnk) 117 c~a mutual energy, 117 bg,q 117

fr 118 C(·; bg,q) Besov capacity, 118

C(·; f!,q) Lizorkin-Triebel capacity, 118

mI! atomic nonlinear potential, 119 a,p m:,p,q atomic nonlinear potential, 120

iink = ţt(Qnk) 121 c(u, F) local degree of approximation,

122 ca operator, 122

Chapter 5

B, balI, 129 A~), A~OO) Hausdorff content (Hausdorff

capacity), 132 Ah, Aa Hausdorff measure, 132 Q, QII meshes of cubes, 136

.c = (h}g" 142 Ee Cantor set, 143

Chapter 6

fiE trace, 158 1 quasicontinuous representative, 158 b(E) set where E is thick, 165, 176 e(E) set where E is thin, 165, 176 ba.p(E) set where E is (a, p)-thick, 166 ea,p(E) set where E is (a, p)-thin, 166

Ca,p (a, p)-capacity 167 E fine closure of E, 176 BV functions of bounded variation, 181

BVa 181

List of Symbols 365

Chapter 7

ţt E restriction of ţt, 191 Bq(ţt) 192 ra,p(') norm, 199 Ku convex set in LI', 199 LP(Ca.p ) space of quasicontinuous

functions, 200 C1,p(K; il) condenser capacity, 209

~tp(K; il) condenser capacity, 209 c:..m,p(F; il) condenser capacity, 209 W,.,P(il) Sobolev space, 209

K(ţt; t) 211 A~OO) dyadic Hausdorff capacity, 212 Ll(A~a) space of quasicontinuous

functions, 212

Chapter 8

fiE, fiE = O trace, 219 1 quasicontinuous representative, 219

Ck,p;6 modified capacity, 223 Ck,p relative capacity, 223 E,. extension operator, 227 llJ ni space of polynomials of degree ~ m,

228

Chapter 9

fiE, fiE = O trace, 234 1 quasicontinuous representative, 234 TrE trace operator, 234 KerTrE kemel of TrE, 234 [f]. average over dyadic balI, 245 Qn meshes of cubes, 246 Qni cubes in Qn, 246 I(Q) side of cube, 246 r Q concentric cube, 246

Qni = 7Qni 246 Q(f) average over Q, 247

Ani 247 Pn(x) 248 Gn 248,252

366 List of Symbols

p(x) 248 r(x) 249 In, b. 249

}..~i 251 p:(x) 251 p~(x) 252 Gn , G~ 252 ro (x) 252 UĂ , U{, u; 263, 264, 272

Chapter 10

L~,P(E) closure of functions with

compact support, 281 Qnk dyadic cube, 282

Qnk = 3Qnk 282 Xnk, Xnk characteristic functions, 282

001 E ideal, 293 s;P, s;P m spaces of polynomials, 293 J m polynomial ideal, 293

s;P /J ni quotient ring, 293 (!y ideal, 294 .c(V) linear hull, 294

Chapter 11

q = p/(p - 1) 305 LZ (E) space of analytic functions, 305 Rq (E) closure of rational functions, 305 L~(E).l 306 Rq(E).l 306 ă Cauchy-Riemann operator, 306 W~'P(E) 309,313

/'(00) 311 Yq(K) analytic capacity, 311 Dq(G), Dq(E) spaces of harmonic

functions, 312 Dq(G).l 313

TrE trace operator, 318 L~'P(E) 318 .c partial differential operator, 319

L1(E) space of solutions to .cu = 0,319 R1(E) space of solutions to .cu = O, 319 R1(E).l, L1(G).l 320

s;p. space of polynomials of degree ~ s, 322

Na,p;a(E; il) polynomial capacity, 322

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

A Selection

219. Duvaut/Lions: Inequalities in Mechanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry 1: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. BerghlLOfstrom: Interpolation Spaces. An Introduction 224. Gilbargffrudinger: Elliptic Partial Differential Equations of Second Order 225. Schtitte: ProofTheory 226. Karoubi: K-Theory. An Introduction 227. GrauertlRemmert: Theorie der Steinschen Răume 228. SegaVKunze: Integrals and Operators 229. Hasse: Number Theory 230. Klingenberg: Lectures on Closed Geodesics 231. Lang: Elliptic Curves. Diophantine Analysis 232. Gihman/Skorohod: The Theory of Stochastic Processes III 233. Stroock/Varadhan: Multidimensional Diffusion Processes 234. Aigner: Combinatorial Theory 235. Dynkin/Yushkevich: Controlled Markov Processes 236. GrauertlRemmert: Theory of Stein Spaces 237. Kothe: Topological Vector Spaces II 238. GrahamIMcGehee: Essays in Commutative Harmonic Analysis 239. Elliott: Probabilistic Number Theory I 240. Elliott: Probabilistic Number Theory II 241. Rudin: Function Theory in the Unit Ball of Cn

242. Huppert/Blackbum: Finite Groups II 243. Huppert/Blackbum: Finite Groups III 244. Kubert/Lang: Modular Units 245. ComfeldIFomin/Sinai: Ergodic Theory 246. NaimarklStem: Theory of Group Representations 247. Suzuki: Group Theory I 248. Suzuki: Group Theory II 249. Chung: Lectures from Markov Processes to Brownian Motion 250. Amold: Geometrical Methods in the Theory of Ordinary Differential Equations 251. ChowlHale: Methods of Bifurcation Theory 252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampere Equations 253. Dwork: Lectures on p-adic Differential Equations 254. Freitag: Siegelsche Modulfunktionen 255. Lang: Complex Multiplication 256. Hormander: The Analysis of Linear Partial Differential Operators I 257. Hormander: The Analysis of Linear Partial Differential Operators II 258. Smoller: Shock Waves and Reaction-Diffusion Equations 259. Duren: Univalent Functions 260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems 261. BoschlGiintzerlRemmert: Non Archimedian Analysis - A System Approach

to Rigid Analytic Geometry 262. Doob: Classical Potential Theory and Its Probabilistic Counterpart 263. Krasnosel'skii'lZabreiko: Geometrical Methods of Nonlinear Analysis 264. Aubin/Cellina: Differential Inclusions 265. GrauertlRemmert: Coherent Analytic Sheaves 266. de Rham: Differentiable Manifolds 267. Arbarello/ComalbalGriffithslHarris: Geometry of Algebraic Curves, VoI. I 268. Arbarello/ComalbalGriffiths/Harris: Geometry of Aigebraic Curves, VoI. II

269. Schapira: Microdifferential Systems in the Complex Domain 270. Scharlau: Quadratic and Hermitian Forms 271. Ellis: Entropy, Large Deviations, and Statistical Mechanics 272. Elliott: Arithmetic Functions and Integer Products 273. Nikol'skii: Treatise on the Shift Operator 274. Hormander: The Analysis of Linear Partial Differential Operators III 275. Hormander: The Analysis of Linear Partial Differential Operators IV 276. Liggett: Interacting Particle Systems 277. Fulton/Lang: Riemann-Roch Algebra 278. BarrlWells: Toposes, Triples and Theories 279. BishoplBridges: Constructive Analysis 280. Neukirch: Class Field Theory 281. Chandrasekharan: Elliptic Functions 282. Lelong/Gruman: Entire Functions of Several Complex Variables 283. Kodaira: Complex Manifolds and Deformation of Complex Structures 284. Finn: Equilibrium Capillary Surfaces 285. BuragolZalgaller: Geometric Inequalities 286. Andrianaov: Quadratic Forms and Hecke Operators 287. Maskit: Kleinian Groups 288. JacodlShiryaev: Limit Theorems for Stochastic Processes 289. Manin: Gauge Fie1d Theory and Complex Geometry 290. Conway/Sloane: Sphere Packings, Lattices and Groups 291. HahnlO'Meara: The Classical Groups and K-Theory 292. KashiwaralSchapira: Sheaves on Manifolds 293. RevuzIYor: Continuous Martingales and Brownian Motion 294. Knus: Quadratic and Hermitian Forms over Rings 295. DierkeslHildebrandtlKiisterlWohlrab: Minimal Surfaces I 296. DierkeslHildebrandtIKiisterlWohlrab: Minimal Surfaces II 297. PasturlFigotin: Spectra of Random and Almost-Periodic Operators 298. BerlinelGetzlerNergne: Heat Kernels and Dirac Operators 299. Pommerenke: Boundary Behaviour of Conformal Maps 300. Orlikfferao: Arrangements of Hyperplanes 301. Loday: Cyclic Homology 302. LangelBirkenhake: Complex Abelian Varieties 303. DeVorelLorentz: Constructive Approximation 304. Lorentz/v. GolitschekIMakovoz: Construcitve Approximation. Advanced Problems 305. Hiriart-UrrutylLemarechal: Convex Analysis and Minimization Algorithms 1.

Fundarnentals 306. Hiriart-UrrutylLemarechal: Convex Analysis and Minimization Algorithms II.

Advanced Theory and Bundle Methods 307. Schwarz: Quantum Field Theory and Topology 308. Schwarz: Topology for Physicists 309. AdemIMilgram: Cohomology of Finite Groups 310. GiaquintaIHildebrandt: Calculus ofVariations 1: The Lagrangian Formalism 311. GiaquintaIHildebrandt: Calculus of Variations II: The Hamiltonian Formalism 312. ChunglZhao: From Brownian Motion to SchrOdinger's Equation 313. Malliavin: Stochastic Analysis 314. AdamslHedberg: Function Spaces and Potential Theory 315. Biirgisser/ClausenlShokrollahi: Algebraic Complexity Theory 316. SafflTotik: Logarithmic Potentials with External Fields 317. RockafellarlWets: Variational Analysis 318. Kobayashi: Hyperbolic Complex Spaces 319. BridsonIHaeftiger: Metric Spaces of Non-Positive Curvature 320. KipnislLandim: Scaling Limits of Interacting Particle Systems 321. Grimmett: Percolation 322. Neukirch: Aigebraic Number Theory 323. NeukirchlSchmidtIWingberg: Cohomology of Number Fields 324. Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes