The Experts below are selected from a list of 102 Experts worldwide ranked by ideXlab platform

Abdelmalek Abdesselam - One of the best experts on this subject based on the ideXlab platform.

  • a second quantized kolmogorov chentsov theorem via the operator product expansion
    Communications in Mathematical Physics, 2020
    Co-Authors: Abdelmalek Abdesselam
    Abstract:

    We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of Pointwise Multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer’s theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson’s operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for Pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field.

Lucas Chaffee - One of the best experts on this subject based on the ideXlab platform.

  • commutators of multilinear singular integral operators with Pointwise Multiplication
    2015
    Co-Authors: Lucas Chaffee
    Abstract:

    In this dissertation we further develop the theory of commutators of multilinear singular integral operators with Pointwise Multiplication. Generally speaking, the commutator of two operators is itself an operator that measures the changes which occur when switching the order in which the commuted operators are being applied. They have proven to be significant historically, and can be useful in the study of PDE. Our first main contribution is the completion of the characterization of the space functions with bounded mean oscillation (BMO) in terms of the boundedness of the corresponding commutator in an appropriate set of Lebesgue spaces. It is already known in a variety of settings that a function being in BMO is sufficient to conclude the boundedness of the commutator, we were able to show that this condition is in fact necessary, a long standing open question. Our characterization opened the door for us to obtain our second main result, namely the necessary and sufficient conditions which guarantee the compactness of the commutator of the bilinear singular integral with Pointwise Multiplication in appropriate weighted Lebesugue spaces.

Abdesselam Abdelmalek - One of the best experts on this subject based on the ideXlab platform.

  • A Second-Quantized Kolmogorov-Chentsov Theorem via the Operator Product Expansion
    2019
    Co-Authors: Abdesselam Abdelmalek
    Abstract:

    We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of Pointwise Multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer's theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson's operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for Pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field.Comment: 50 pages, 4 figures, this is the final version of the articl

B Pavlovic - One of the best experts on this subject based on the ideXlab platform.

  • automatic continuity of lipschitz algebras
    Journal of Functional Analysis, 1995
    Co-Authors: B Pavlovic
    Abstract:

    Abstract For a compact metric space ( K , d ), α ∈ (0,1] and f ∈ C ( K ), let p α ( f ) = sup{| f ( t ) − f ( s )|/ d ( t , s ) α : t , s ∈ K }. The set Lip α ( K , d ) = { f ∈ C ( K ): p α ( f ) f || α = | f | K + p α ( f ) is a Banach function algebra under Pointwise Multiplication. The subset lip α ( K , D ) = { f ∈ Lip α ( K , d ) : | f ( t ) − f ( s )|/ d ( t , s ) α → 0 as d ( t , s ) → 0} is a closed subalgebra of Lip α ( K , d ). For 0 α ( K , d ) ⊇ Lip β ( K , d ) ⊇ lip β ( K , d ) and so they form a one parameter family of algebras ordered by inclusion. Let A α = Lip α ( K , d ) or lip α ( K , d ). For α, β ∈ (0, 1), the relationship between the ideals of A α and A β is examined, and important inclusions of different such ideals derived. This enables us to establish automatic continuity properties of Lipschitz algebras. A homomorphism v : A → B , B a Banach algebra, is said to be eventually continuous if ∃β ≥α such that v | A β is continuous for the ||·|| β -norm. First, it is shown that in Lipschitz algebras, when α ∈ (0, 1 2 ), the eventual continuity is equivalent to nilpotency of the separating ideal in the range algebra. This is used to prove that if α ∈ (0, 1 2 ), and v is eventually continuous, then v is continuous on A 2α + γ for all γ ∈ (0, 1 − 2α). It is also shown that in these algebras the prime ideals containing a given prime ideal form a chain. All these results are then used to prove that for α ∈ (0, 1 2 ) every epimorphism from A α is eventually continuous. This research extends work of Bade, Curties, and Laursen, who considered these same questions for C n ([0, 1]).

Jean De Canni~re - One of the best experts on this subject based on the ideXlab platform.