On Necessary And Sufficient Conditions For Lp Estimates Of Riesz Transforms Associated To Elliptic Operators On Rn And Related Estimates
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| Author | : Pascal Auscher |
| Publisher | : American Mathematical Society(RI) |
| Total Pages | : 102 |
| Release | : 2014-09-11 |
| Genre | : MATHEMATICS |
| ISBN | : 9781470404758 |
This memoir focuses on $Lp$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. The author introduces four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the $Lp$ estimates. It appears that the case $p2$ which is new. The author thus recovers in a unified and coherent way many $Lp$ estimates and gives further applications. The key tools from harmonic analysis are two criteria for $Lp$ boundedness, one for $p2$ but in ranges different from the usual intervals $(1,2)$ and $(2, \infty)$.
| Author | : Pascal Auscher |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 102 |
| Release | : 2007 |
| Genre | : Mathematics |
| ISBN | : 0821839411 |
This memoir focuses on $Lp$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. The author introduces four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the $Lp$ estimates. It appears that the case $p2$ which is new. The author thus recovers in a unified and coherent way many $Lp$ estimates and gives further applications. The key tools from harmonic analysis are two criteria for $Lp$ boundedness, one for $p2$ but in ranges different from the usual intervals $(1,2)$ and $(2,\infty)$.
| Author | : Cédric Arhancet |
| Publisher | : Springer Nature |
| Total Pages | : 288 |
| Release | : 2022-05-05 |
| Genre | : Mathematics |
| ISBN | : 3030990117 |
This book on recent research in noncommutative harmonic analysis treats the Lp boundedness of Riesz transforms associated with Markovian semigroups of either Fourier multipliers on non-abelian groups or Schur multipliers. The detailed study of these objects is then continued with a proof of the boundedness of the holomorphic functional calculus for Hodge–Dirac operators, thereby answering a question of Junge, Mei and Parcet, and presenting a new functional analytic approach which makes it possible to further explore the connection with noncommutative geometry. These Lp operations are then shown to yield new examples of quantum compact metric spaces and spectral triples. The theory described in this book has at its foundation one of the great discoveries in analysis of the twentieth century: the continuity of the Hilbert and Riesz transforms on Lp. In the works of Lust-Piquard (1998) and Junge, Mei and Parcet (2018), it became apparent that these Lp operations can be formulated on Lp spaces associated with groups. Continuing these lines of research, the book provides a self-contained introduction to the requisite noncommutative background. Covering an active and exciting topic which has numerous connections with recent developments in noncommutative harmonic analysis, the book will be of interest both to experts in no-commutative Lp spaces and analysts interested in the construction of Riesz transforms and Hodge–Dirac operators.
| Author | : Patricio Cifuentes |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 258 |
| Release | : 2010-01-14 |
| Genre | : Mathematics |
| ISBN | : 0821847708 |
This volume contains the Proceedings of the 8th International Conference on Harmonic Analysis and Partial Differential Equations, held in El Escorial, Madrid, Spain, on June 16-20, 2008. Featured in this book are papers by Steve Hoffmann and Carlos Kenig, which are based on two mini-courses given at the conference. These papers present topics of current interest, which assume minimal background from the reader, and represent state-of-the-art research in a useful way for young researchers. Other papers in this volume cover a range of fields in Harmonic Analysis and Partial Differential Equations and, in particular, illustrate well the fruitful interplay between these two fields.
| Author | : Steve Hofmann |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 91 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 0821852388 |
Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.
| Author | : El-Maati Ouhabaz |
| Publisher | : Princeton University Press |
| Total Pages | : 296 |
| Release | : 2009-01-10 |
| Genre | : Mathematics |
| ISBN | : 1400826489 |
This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treats Lp properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics. This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to prove Lp estimates for heat, Schrödinger, and wave type equations. A significant part of the results have been proved during the last decade. The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications. It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.
| Author | : Zhongwei Shen |
| Publisher | : Springer |
| Total Pages | : 295 |
| Release | : 2018-09-04 |
| Genre | : Mathematics |
| ISBN | : 3319912143 |
This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.
| Author | : Gail Letzter |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 104 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 0821841319 |
This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.
| Author | : Sebastian Bechtel |
| Publisher | : Springer Nature |
| Total Pages | : 191 |
| Release | : |
| Genre | : |
| ISBN | : 3031637682 |
| Author | : Charlotte Wahl |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 130 |
| Release | : 2007 |
| Genre | : Index theory |
| ISBN | : 0821839977 |
The author defines and proves a noncommutative generalization of a formula relating the Maslov index of a triple of Lagrangian subspaces of a symplectic vector space to eta-invariants associated to a pair of Lagrangian subspaces. The noncommutative Maslov index, defined for modules over a $C *$-algebra $\mathcal{A}$, is an element in $K_0(\mathcal{A})$. The generalized formula calculates its Chern character in the de Rham homology of certain dense subalgebras of $\mathcal{A}$. The proof is a noncommutative Atiyah-Patodi-Singer index theorem for a particular Dirac operator twisted by an $\mathcal{A}$-vector bundle. The author develops an analytic framework for this type of index problem.
