Convolution-dominated integral operators

Tom 89 / 2010

Gero Fendler, Karlheinz Gröchenig, Michael Leinert Banach Center Publications 89 (2010), 121-127 MSC: Primary 47B35; Secondary 43A20. DOI: 10.4064/bc89-0-6

Streszczenie

For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution-dominated operators on $L^{2}(G)$, where an operator $A:L^2(G)\to L^2(G)$ is called convolution-dominated if there exists $a\in L^1(G)$ such that for all $f \in L^2(G)$ $$ |Af(x)| \leq a \star |f|(x), \quad\ \hbox{for almost all } x\in G.\tag{1} $$ The case of discrete groups was treated in previous publications \cite{fgl08a, fgl08}. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the products are restricted to those given by multiplication with left uniformly continuous functions. This algebra, $CD_{reg}(G)$, is canonically isomorphic to a twisted $L^{1}$-algebra. For amenable $G$ that is rigidly symmetric as a discrete group we show the following result: An element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if and only if it is invertible as a bounded operator on $L^2(G)$. This report is about work in progress. Complete details and further results will be given in a paper still in preparation.

Autorzy

  • Gero FendlerFakultät für Mathematik
    Universität Wien
    Nordbergstrasse 15
    A-1090 Wien, Austria
    e-mail
  • Karlheinz GröchenigFakultät für Mathematik
    Universität Wien
    Nordbergstrasse 15
    A-1090 Wien, Austria
    e-mail
  • Michael LeinertInstitut für Angewandte Mathematik
    Fakultät für Mathematik
    Im Neuenheimer Feld 288
    D-69120 Heidelberg, Germany
    e-mail

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