Table Of Content

Hörmander Operators TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk Hörmander Operators Marco Bramanti Politecnico di Milano, Italy Luca Brandolini Università degli Studi di Bergamo, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Control Number: 2022047054 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. HÖRMANDER OPERATORS Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-126-168-8 (hardcover) ISBN 978-981-126-169-5 (ebook for institutions) ISBN 978-981-126-170-1 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13006#t=suppl Printed in Singapore To our families and parents. TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk Contents Foreword xi Introduction xiii 0.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 0.2 Scope and structure of the book . . . . . . . . . . . . . . . . . . . xviii 0.3 Why study H¨ormander operators? . . . . . . . . . . . . . . . . . xxii 1. Basic geometry of vector fields 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Exponentials and commutators of vector fields. . . . . . . . . . . 2 1.3 Lie algebras, H¨ormander’s condition, H¨ormander operators. . . . 8 1.4 The control distance . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 The weighted control distance . . . . . . . . . . . . . . . . . . . . 21 1.6 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Other properties related to connectivity . . . . . . . . . . . . . . 35 1.8 Maximum principles for degenerate elliptic operators . . . . . . . 37 1.9 Propagation of maxima and strong maximum principle for sum of squares operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.10 Propagation of maxima for operators with drift . . . . . . . . . . 48 1.11 Some examples of explicit computations with the control distance 56 1.12 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2. Function spaces defined by systems of vector fields 67 2.1 Sobolev spaces induced by vector fields . . . . . . . . . . . . . . . 67 2.2 H¨older spaces induced by H¨ormander vector fields. . . . . . . . . 80 2.3 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3. Homogeneous groups in RN 93 3.1 Homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . 94 vii viii Ho¨rmander operators 3.2 Homogeneous Lie algebras of invariant vector fields on a homogeneous group . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3 Exponential maps on a homogeneous group . . . . . . . . . . . . 116 3.4 Convolution and mollifiers on a homogeneous group. . . . . . . . 118 3.5 Homogeneous stratified Lie groups and Lie algebras, and their control distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.6 Connectivity matters and Poincar´e inequality on stratified groups 128 3.7 Weak solutions to Dirichlet problems for divergence form equations structured on vector fields . . . . . . . . . . . . . . . . 132 3.8 Homogeneous stratified Lie algebras and Lie groups of type II . . 135 3.9 Distributions on homogeneous groups. . . . . . . . . . . . . . . . 141 3.10 Examples of homogeneous groups and homogeneous H¨ormander operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.11 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4. Hypoellipticity of sublaplacians on Carnot groups 153 4.1 Introduction, statement of the main results and strategy of the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2 Notation and preliminary facts about Sobolev spaces and finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.3 Regularity estimates for the canonical sublaplacian . . . . . . . . 163 4.4 Hypoellipticity of the canonical sublaplacian . . . . . . . . . . . . 174 4.5 General sublaplacians and uniform estimates . . . . . . . . . . . 184 4.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5. Hypoellipticity of general H¨ormander operators 191 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.2 The Fourier transform on the Schwartz space S(Rn) and on tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . 192 5.3 Fractional order Sobolev spaces . . . . . . . . . . . . . . . . . . . 195 5.4 Some classes of operators on S(Rn) . . . . . . . . . . . . . . . . . 199 5.5 Subelliptic estimates . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.6 Localized subelliptic estimate . . . . . . . . . . . . . . . . . . . . 229 5.7 Hypoellipticity of H¨ormander operators . . . . . . . . . . . . . . 232 5.8 Uniform subelliptic estimates . . . . . . . . . . . . . . . . . . . . 238 5.9 Some applications of the subelliptic estimates . . . . . . . . . . . 242 5.10 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6. Fundamental solutions of H¨ormander operators 247 6.1 Fundamental solutions and solvability of general H¨ormander operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.2 Homogeneous H¨ormander operators . . . . . . . . . . . . . . . . . 254 Contents ix 6.3 Existence of a global homogeneous fundamental solution and uniform estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6.4 Properties of the global fundamental solution . . . . . . . . . . . 268 6.5 Some explicit examples of fundamental solutions on homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7. Real analysis and singular integrals in locally doubling metric spaces 291 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.2 Locally doubling metric spaces . . . . . . . . . . . . . . . . . . . 295 7.3 Localized kernels of singular and fractional integrals . . . . . . . 298 7.4 Singular and fractional integrals on H¨older spaces . . . . . . . . . 301 7.5 L2 continuity of singular integrals via continuity on Cα . . . . . 309 7.6 Local maximal function and fractional integrals on Lp spaces . . 312 7.7 Calder´on-Zygmund theory in locally doubling metric spaces . . . 318 7.8 Integral characterization of H¨older continuity . . . . . . . . . . . 327 7.9 Some geometric results . . . . . . . . . . . . . . . . . . . . . . . . 331 7.10 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 8. Sobolev and H¨older estimates for H¨ormander operators on groups 337 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 8.2 Homogeneous kernels on G, fractional integrals and Sobolev embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 8.3 Singular integrals associated to homogeneous kernels of type 0 . . 354 8.4 Global Sobolev estimates. . . . . . . . . . . . . . . . . . . . . . . 360 8.5 Local Sobolev estimates . . . . . . . . . . . . . . . . . . . . . . . 370 8.6 H¨older estimates for solutions of Lu=f . . . . . . . . . . . . . . 378 8.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9. More geometry of vector fields: metric balls and equivalent distances 399 9.1 Introduction and statement of the main results . . . . . . . . . . 399 9.2 Dependence of the constants . . . . . . . . . . . . . . . . . . . . . 406 9.3 The Baker-Campbell-Hausdorff formula . . . . . . . . . . . . . . 408 9.4 Suboptimal bases and their properties . . . . . . . . . . . . . . . 420 9.5 Structure of metric balls . . . . . . . . . . . . . . . . . . . . . . . 437 9.6 Local equivalence of the distances d,d∗ . . . . . . . . . . . . . . . 458 9.7 Segment properties and the global doubling condition . . . . . . 462 9.8 Proof of the BCH formula for formal series and other consequences 466 9.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 10. Lifting and approximation 479 10.1 Motivation and statement of the main results . . . . . . . . . . . 479

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Hörmander Operators TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk Hörmander Operators Marco Bramanti Politecnico di Milano, Italy Luca Brandolini Università degli Studi di Bergamo, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Control Number: 2022047054 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. HÖRMANDER OPERATORS Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-126-168-8 (hardcover) ISBN 978-981-126-169-5 (ebook for institutions) ISBN 978-981-126-170-1 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13006#t=suppl Printed in Singapore To our families and parents. TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk Contents Foreword xi Introduction xiii 0.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 0.2 Scope and structure of the book . . . . . . . . . . . . . . . . . . . xviii 0.3 Why study H¨ormander operators? . . . . . . . . . . . . . . . . . xxii 1. Basic geometry of vector fields 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Exponentials and commutators of vector fields. . . . . . . . . . . 2 1.3 Lie algebras, H¨ormander’s condition, H¨ormander operators. . . . 8 1.4 The control distance . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 The weighted control distance . . . . . . . . . . . . . . . . . . . . 21 1.6 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Other properties related to connectivity . . . . . . . . . . . . . . 35 1.8 Maximum principles for degenerate elliptic operators . . . . . . . 37 1.9 Propagation of maxima and strong maximum principle for sum of squares operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.10 Propagation of maxima for operators with drift . . . . . . . . . . 48 1.11 Some examples of explicit computations with the control distance 56 1.12 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2. Function spaces defined by systems of vector fields 67 2.1 Sobolev spaces induced by vector fields . . . . . . . . . . . . . . . 67 2.2 H¨older spaces induced by H¨ormander vector fields. . . . . . . . . 80 2.3 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3. Homogeneous groups in RN 93 3.1 Homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . 94 vii viii Ho¨rmander operators 3.2 Homogeneous Lie algebras of invariant vector fields on a homogeneous group . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3 Exponential maps on a homogeneous group . . . . . . . . . . . . 116 3.4 Convolution and mollifiers on a homogeneous group. . . . . . . . 118 3.5 Homogeneous stratified Lie groups and Lie algebras, and their control distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.6 Connectivity matters and Poincar´e inequality on stratified groups 128 3.7 Weak solutions to Dirichlet problems for divergence form equations structured on vector fields . . . . . . . . . . . . . . . . 132 3.8 Homogeneous stratified Lie algebras and Lie groups of type II . . 135 3.9 Distributions on homogeneous groups. . . . . . . . . . . . . . . . 141 3.10 Examples of homogeneous groups and homogeneous H¨ormander operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.11 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4. Hypoellipticity of sublaplacians on Carnot groups 153 4.1 Introduction, statement of the main results and strategy of the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2 Notation and preliminary facts about Sobolev spaces and finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.3 Regularity estimates for the canonical sublaplacian . . . . . . . . 163 4.4 Hypoellipticity of the canonical sublaplacian . . . . . . . . . . . . 174 4.5 General sublaplacians and uniform estimates . . . . . . . . . . . 184 4.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5. Hypoellipticity of general H¨ormander operators 191 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.2 The Fourier transform on the Schwartz space S(Rn) and on tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . 192 5.3 Fractional order Sobolev spaces . . . . . . . . . . . . . . . . . . . 195 5.4 Some classes of operators on S(Rn) . . . . . . . . . . . . . . . . . 199 5.5 Subelliptic estimates . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.6 Localized subelliptic estimate . . . . . . . . . . . . . . . . . . . . 229 5.7 Hypoellipticity of H¨ormander operators . . . . . . . . . . . . . . 232 5.8 Uniform subelliptic estimates . . . . . . . . . . . . . . . . . . . . 238 5.9 Some applications of the subelliptic estimates . . . . . . . . . . . 242 5.10 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6. Fundamental solutions of H¨ormander operators 247 6.1 Fundamental solutions and solvability of general H¨ormander operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.2 Homogeneous H¨ormander operators . . . . . . . . . . . . . . . . . 254 Contents ix 6.3 Existence of a global homogeneous fundamental solution and uniform estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6.4 Properties of the global fundamental solution . . . . . . . . . . . 268 6.5 Some explicit examples of fundamental solutions on homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7. Real analysis and singular integrals in locally doubling metric spaces 291 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.2 Locally doubling metric spaces . . . . . . . . . . . . . . . . . . . 295 7.3 Localized kernels of singular and fractional integrals . . . . . . . 298 7.4 Singular and fractional integrals on H¨older spaces . . . . . . . . . 301 7.5 L2 continuity of singular integrals via continuity on Cα . . . . . 309 7.6 Local maximal function and fractional integrals on Lp spaces . . 312 7.7 Calder´on-Zygmund theory in locally doubling metric spaces . . . 318 7.8 Integral characterization of H¨older continuity . . . . . . . . . . . 327 7.9 Some geometric results . . . . . . . . . . . . . . . . . . . . . . . . 331 7.10 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 8. Sobolev and H¨older estimates for H¨ormander operators on groups 337 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 8.2 Homogeneous kernels on G, fractional integrals and Sobolev embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 8.3 Singular integrals associated to homogeneous kernels of type 0 . . 354 8.4 Global Sobolev estimates. . . . . . . . . . . . . . . . . . . . . . . 360 8.5 Local Sobolev estimates . . . . . . . . . . . . . . . . . . . . . . . 370 8.6 H¨older estimates for solutions of Lu=f . . . . . . . . . . . . . . 378 8.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9. More geometry of vector fields: metric balls and equivalent distances 399 9.1 Introduction and statement of the main results . . . . . . . . . . 399 9.2 Dependence of the constants . . . . . . . . . . . . . . . . . . . . . 406 9.3 The Baker-Campbell-Hausdorff formula . . . . . . . . . . . . . . 408 9.4 Suboptimal bases and their properties . . . . . . . . . . . . . . . 420 9.5 Structure of metric balls . . . . . . . . . . . . . . . . . . . . . . . 437 9.6 Local equivalence of the distances d,d∗ . . . . . . . . . . . . . . . 458 9.7 Segment properties and the global doubling condition . . . . . . 462 9.8 Proof of the BCH formula for formal series and other consequences 466 9.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 10. Lifting and approximation 479 10.1 Motivation and statement of the main results . . . . . . . . . . . 479

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