Holdings: Maximal subellipticity

Maximal subellipticity:

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Bibliographic Details
Main Author:	Street, Brian 1981- (Author)
Format:	Book
Language:	English
Published:	Berlin ; Boston De Gruyter [2023]
Series:	De Gruyter Studies in Mathematics Volume 93
Subjects:	subelliptisch sub-riemann volstandig nichtlinear subelliptic, hypoelliptic, degenerate elliptic, sub-Riemannian, fully nonlinear subelliptisch; sub-riemann; volstandig nichtlinear subelliptic; hypoelliptic; degenerate elliptic; sub-Riemannian; fully nonlinear
Online Access:	https://www.degruyter.com/isbn/9783111085173 Inhaltsverzeichnis
Physical Description:	X, 756 Seiten 24 cm x 17 cm, 1353 g
ISBN:	9783111085173 3111085171

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653			\|a subelliptisch
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653			\|a subelliptic, hypoelliptic, degenerate elliptic, sub-Riemannian, fully nonlinear
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adam_txt	CONTENTS 1 INTRODUCTION - 1 1.1 BASIC DEFINITIONS - 2 1.1.1 LINEAR MAXIMALLY SUBELLIPTIC OPERATORS - 5 1.1.2 NONLINEAR MAXIMALLY SUBELLIPTIC OPERATORS - 9 1.2 1.3 1.4 1.5 1.6 1.7 BACKGROUND: ELLIPTIC THEORY - 12 CARNOT-CARATHEODORY GEOMETRY - 14 SINGULAR INTEGRALS - 15 FUNCTION SPACES - 17 LINEAR THEORY - 18 FULLY NONLINEAR THEORY - 20 1.7.1 THE INVERSE FUNCTION THEOREM - 21 1.8 1.9 MULTI-PARAMETER THEORY - 24 THE MAIN TOOL: SCALING - 27 1.9.1 SCALING FOR ELLIPTIC OPERATORS - 28 1.9.2 SCALING FOR MAXIMALLY SUBELLIPTIC OPERATORS - 28 1.10 OUTLINE - 29 2 ELLIPTICITY - 31 2.1 2.2 SOME BASIC FUNCTION SPACES - 32 PSEUDO-DIFFERENTIAL OPERATORS AND THE FOURIER TRANSFORM - 35 2.2.1 PSEUDO-DIFFERENTIAL OPERATORS WITHOUT THE FOURIER TRANSFORM - 37 2.2.2 ELLIPTIC OPERATORS AND PSEUDO-DIFFERENTIAL OPERATORS - 42 2.3 SINGULAR INTEGRAL OPERATORS - 47 2.3.1 LOCAL, MATRIX-VALUED OPERATORS - 58 2.4 2.5 BESOV AND TRIEBEL-LIZORKIN SPACES - 59 ZYGMUND-HBLDER SPACES - 70 2.5.1 DECOMPOSITION INTO SMOOTH FUNCTIONS - 76 2.6 2.7 2.8 COMPOSITIONS AND PRODUCT ZYGMUND-HDLDER SPACES - 78 LINEAR ELLIPTIC OPERATORS - 87 NONLINEAR ELLIPTIC EQUATIONS - 90 2.8.1 REDUCTION I - 93 2.8.2 REDUCTION II - 96 2.8.3 COMPLETION OF THE PROOF - 98 2.8.4 WEIGHTED ESTIMATES NEAR THE BOUNDARY - 106 2.9 FURTHER READING AND REFERENCES - 111 3 VECTOR FIELDS AND CARNOT-CARATHEODORY GEOMETRY - 114 3.1 MANIFOLDS - 114 3.1.1 THE EXPONENTIAL MAP - 116 3.1.2 THE BAKER-CAMPBELL-HAUSDORFF FORMULA - 118 VI - CONTENTS 3.1.3 THE FROBENIUS THEOREM - 119 3.2 THE UNIT SCALE - 121 3.3 SCALING AND HORMANDER VECTOR FIELDS - 123 3.3.1 SCALING AND MAXIMAL SUBELLIPTICITY - 127 3.4 FINITELY GENERATED SETTING - 129 3.5 MULTI-PARAMETER CARNOT-CARATHEODORY GEOMETRY - 135 3.5.1 AN IMPORTANT SPECIAL CASE: HORMANDER ' S CONDITION - 137 3.5.2 DROPPING PARAMETERS - 138 3.6 THE QUANTITATIVE COORDINATE SYSTEM - 139 3.7 PROOFS OF SCALING RESULTS - 142 3.8 APPROXIMATELY COMMUTING VECTOR FIELDS - 148 3.9 INTEGRATING OVER LEAVES - 155 3.10 INTEGRALS AND APPROXIMATELY COMMUTING VECTOR FIELDS - 158 3.11 MAXIMALFUNCTIONS - 166 3.11.1 THE FINITELY GENERATED SETTING - 167 3.11.1.1 A RESULT ON THE UNIT BALL - 170 3.11.1.2 PROOF OF THEOREM 3.11.2 - 172 3.11.2 THE HORMANDER SETTING - 174 3.12 APPROXIMATELY COMMUTING PARTIAL DIFFERENTIAL OPERATORS - 177 3.13 FILTERED MODULES OF VECTOR FIELDS - 186 3.14 CONTROL OF VECTOR FIELDS - 188 3.14.1 FURTHER COMMENTS ON EQUIVALENCE - 190 3.15 THE MAIN MULTI-PARAMETER SETTING - 194 3.15.1 THE MULTI-PARAMETER UNIT SCALE - 194 3.15.2 SCALING - 195 3.15.3 QUANTITATIVE SCALING - 198 3.16 FURTHER READING AND REFERENCES - 203 4 PSEUDO-DIFFERENTIAL OPERATORS - 205 4.1 SYMBOLS OF PSEUDO-DIFFERENTIAL OPERATORS - 206 4.1.1 CONNECTION WITH STANDARD PSEUDO-DIFFERENTIAL OPERATORS - 208 4.1.2 LITTLEWOOD-PALEY DECOMPOSITION OF SYMBOLS - 209 4.2 THE EXPONENTIAL MAP - 215 4.3 LITTLEWOOD-PALEY DECOMPOSITIONS - 223 4.3.1 LEBESGUE SPACE BOUNDS - 231 4.3.2 ADJOINTS - 234 4.3.3 PROOF OF THE LITTLEWOOD-PALEY DECOMPOSITION - 236 4.4 ADDING PARAMETERS - 238 4.5 THE SUB-LAPLACIAN - 239 4.5.1 THE SUB-LAPLACIAN IS SUBELLIPTIC - 240 4.5.2 HOMOGENEITY - 250 4.5.3 NILPOTENT LIE GROUPS - 254 CONTENTS - VII 4.6 FURTHER READING AND REFERENCES - 265 4.5.4 4.5.5 4.5.6 4.5.7 FREE NILPOTENT LIE ALGEBRAS - 256 THE CALCULUS - 257 THE PARAMETRIX - 262 LIMITATIONS OF THE LIFTING PROCEDURE - 264 5 SINGULAR INTEGRALS - 267 5.1 THE THREE SETTINGS - 267 5.1.1 THE SINGLE-PARAMETER SETTING - 267 5.1.2 THE MULTI-PARAMETER HORMANDER SETTING - 268 5.1.3 THE GENERAL MULTI-PARAMETER SETTING - 269 5.2 THE ALGEBRAS OF SINGULAR INTEGRALS - 270 5.2.1 THE SINGLE-PARAMETER SETTING - 271 5.2.2 THE MULTI-PARAMETER HORMANDER SETTING - 274 5.2.3 THE GENERAL MULTI-PARAMETER SETTING - 276 5.3 NOTATION - 279 5.4 PRE-ELEMENTARY OPERATORS - 280 5.5 ELEMENTARY OPERATORS - 289 5.6 PSEUDO-DIFFERENTIAL OPERATORS - 310 5.7 EQUIVALENCE OF THE DEFINITIONS - 316 5.8 BASIC PROPERTIES - 329 5.8.1 PSEUDO-LOCALITY - 334 5.8.2 STANDARD PSEUDO-DIFFERENTIAL OPERATORS - 338 5.9 AN IMPORTANT SUBALGEBRA - 340 5.9.1 STANDARD PSEUDO-DIFFERENTIAL OPERATORS - 353 5.10 L P BOUNDS IN THE SINGLE-PARAMETER CASE - 355 5.10.1 A UNIFORM RESULT - 358 5.10.2 BEYOND HORMANDER' S CONDITION - 359 5.11 PARAMETRICES - 369 5.11.1 PARAMETRICES VIA HEAT EQUATIONS - 370 5.11.2 PARAMETRICES AND OTHER GEOMETRIES - 378 5.12 SPECTRAL MULTIPLIERS - 387 5.13 FURTHER READING AND REFERENCES - 389 6 BESOV AND TRIEBEL-LIZORKIN SPACES - 393 6.1 INFORMAL DESCRIPTION OF THE NORMS - 394 6.2 THE SINGLE-PARAMETER SPACES - 396 6.3 THE MULTI-PARAMETER SPACES - 400 6.4 THE MAIN ESTIMATE - 402 6.5 SOME BASIC PROPERTIES - 408 6.5.1 SMOOTH FUNCTIONS ARE DENSE - 422 6.6 THE SINGLE-PARAMETER SPACES, REVISITED - 425 VIII - CONTENTS 6.6.1 THE CLASSICAL SPACES - 427 6.6.2 COMPARING SINGLE-PARAMETER SPACES - 431 6.6.3 BOUNDEDNESS OF OPERATORS IN 9 - 439 6.7 TRADING DERIVATIVES - 445 6.7.1 SHARPNESS - 448 6.8 ADDING PARAMETERS - 458 6.9 SOBOLEV SPACES - 470 6.10 DISTRIBUTIONS OF FINITE NORM - 472 6.11 AN EXPLICIT CHOICE OF NORM - 475 6.11.1 THE UNIT SCALE - 479 6.11.2 SCALING - 479 6.12 A SPECTRAL DEFINITION - 481 6.13 FURTHER QUESTIONS - 482 6.14 FURTHER READING AND REFERENCES - 483 7 ZYGMUND-HDLDER SPACES - 486 7.1 THE NORM - 487 7.2 DECOMPOSITION INTO SMOOTH FUNCTIONS - 490 7.3 BOUNDS OF SOME SUMS - 492 7.4 ALGEBRA - 494 7.5 COMPOSITIONS - 496 7.5.1 PROPERTIES OF THE PRODUCT ZYGMUND-HBLDER SPACES - 499 7.5.2 PROOF OF THE COMPOSITION THEOREM - 501 7.5.3 THE CLASSICAL PRODUCT ZYGMUND-HBLDER SPACES - 513 7.6 ADDING PARAMETERS - 514 7.7 DIFFERENCE CHARACTERIZATION - 515 7.7.1 HOLDER SPACES - 530 8 LINEAR MAXIMALLY SUBELLIPTIC OPERATORS - 534 8.1 THE MAIN RESULT - 534 8.2 FURTHER REGULARITY PROPERTIES - 537 8.2.1 SINGLE-PARAMETER FUNCTION SPACES - 537 8.2.2 STANDARD FUNCTION SPACES - 541 8.2.3 MULTI-PARAMETER FUNCTION SPACES - 546 8.3 A PRIORI ESTIMATES - 551 8.3.1 SOME PRELIMINARY ESTIMATES - 552 8.3.2 SUBELLIPTIC ESTIMATES - 554 8.3.3 EXPONENTIAL ESTIMATES - 566 8.4 HEAT EQUATIONS - 579 8.4.1 STEP I: THE UNIT SCALE - 581 8.4.1.1 ON DIAGONAL BOUNDS - 581 8.4.1.2 OFF-DIAGONAL BOUNDS - 582 CONTENTS - IX 8.4.2 STEP II: A SINGLE POINT AND SCALE - 583 8.4.2.1 ON-DIAGONAL BOUNDS - 585 8.4.2.2 OFF-DIAGONAL BOUNDS - 587 8.4.3 STEP III: ALL SCALES - 588 8.5 PROOF OF THE MAIN RESULT - 598 8.6 VECTOR BUNDLES - 604 8.7 QUANTITATIVE REGULARITY ESTIMATES - 608 8.7.1 THE UNIT SCALE - 608 8.7.2 A SMALL SCALE - 610 8.8 REPRESENTATION THEORY AND ROCKLAND ' S CONDITION - 613 8.9 POSITIVE DEFINITE FORMS - 614 8.10 FURTHER READING AND REFERENCES - 616 9 NONLINEAR MAXIMALLY SUBELLIPTIC EQUATIONS - 619 9.1 MAIN QUALITATIVE RESULTS - 619 9.1.1 SINGLE-PARAMETER RESULTS - 620 9.1.1.1 QUALITATIVE SCHAUDER ESTIMATES - 622 9.1.2 MULTI-PARAMETER RESULTS - 624 9.1.2.1 QUALITATIVE SCHAUDER ESTIMATES - 627 9.2 MAIN QUANTITATIVE RESULT - 627 9.2.1 VECTOR FIELDS AND NORMS - 630 9.2.2 BUMP FUNCTIONS - 632 9.2.3 SCALED ESTIMATES - 640 9.2.4 STEP I: PERTURBATION OF A LINEAR OPERATOR - 646 9.2.5 STEP II: A SIMPLER FORM - 656 9.2.6 SCALED DECOMPOSITIONS OF ZYGMUND-HBLDER FUNCTIONS - 660 9.2.7 STEP III: COMPLETION OF THE PROOF - 662 9.3 WEIGHTED ESTIMATES NEAR THE BOUNDARY - 675 9.3.1 WEIGHTED SCHAUDER ESTIMATES NEAR THE BOUNDARY - 684 9.4 EXAMPLES - 685 9.4.1 SECOND-ORDER EQUATIONS - 685 9.4.2 THE MONGE-AMPERE EQUATION - 687 9.4.3 HIGHER-ORDER MONGE-AMPERE EQUATIONS - 689 9.4.4 HIGHER-ORDER EQUATIONS - 691 9.5 FURTHER READING AND REFERENCES - 693 A CANONICAL COORDINATES - 695 A.1 BASIC NOTATION - 695 A.2 THE MAIN RESULTS - 696 A.2.1 MORE ON THE ASSUMPTIONS - 698 A.2.2 DENSITIES - 701 A.3 QUALITATIVE CONSEQUENCES - 703 X - CONTENTS A.4 PROOFS - 707 A.4.1 AN ODE - 707 A.4.1 .1 DERIVATION OF THE ODE - 707 A.4.1 .2 EXISTENCE, UNIQUENESS, AND REGULARITY - 710 A.4.2 A.4.3 AN INVERSE FUNCTION THEOREM - 716 PROOF OF THE MAIN RESULT - 719 A.4.3.1 LINEARLY INDEPENDENT - 719 A.4.3.2 LINEARLY DEPENDENT - 724 A.4.4 A.4.5 DENSITIES - 732 PROOF OF THEOREM 3.6.5 - 738 A.5 FURTHER READING AND REFERENCES - 738 BIBLIOGRAPHY - 741 SYMBOL INDEX - 753 INDEX - 755
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spelling	Street, Brian 1981- Verfasser (DE-588)1074231368 aut Maximal subellipticity Brian Street Berlin ; Boston De Gruyter [2023] X, 756 Seiten 24 cm x 17 cm, 1353 g txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematics Volume 93 subelliptisch sub-riemann volstandig nichtlinear subelliptic, hypoelliptic, degenerate elliptic, sub-Riemannian, fully nonlinear subelliptisch; sub-riemann; volstandig nichtlinear subelliptic; hypoelliptic; degenerate elliptic; sub-Riemannian; fully nonlinear Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Online-Ausgabe, EPUB 978-3-11-108594-4 Erscheint auch als Online-Ausgabe, PDF 978-3-11-108564-7 De Gruyter Studies in Mathematics Volume 93 (DE-604)BV000005407 93 X:MVB https://www.degruyter.com/isbn/9783111085173 DNB Datenaustausch https://bvbr.bib-bvb.de:443/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034306565&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p vlb 20230210 DE-101 https://d-nb.info/provenance/plan#vlb
spellingShingle	Street, Brian 1981- Maximal subellipticity De Gruyter Studies in Mathematics
title	Maximal subellipticity
title_auth	Maximal subellipticity
title_exact_search	Maximal subellipticity
title_exact_search_txtP	Maximal subellipticity
title_full	Maximal subellipticity Brian Street
title_fullStr	Maximal subellipticity Brian Street
title_full_unstemmed	Maximal subellipticity Brian Street
title_short	Maximal subellipticity
title_sort	maximal subellipticity
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