It is the purpose of this talk to discuss a variety of situations where invariance properties of function spaces under a certain group of operators, specifically time-frequency shifts or dilations, help to derive atomic characterizations, find minimal or maximal spaces, or prove boundedness properties of certain operators.
Aside from the well-known characterization of real Hardy spaces via atomic decompositions([1]) we can mention the work on the Segal algebra (S0(Rd),k ¢ kS0) in the context of Gabor analysis (see [2]), but also the proof of Wiener’s Third Tauberian Theorem (see [3]) for functions of bounded p-means on Rd (Wiener did the case d ˘ 1,p ˘ 2 in his book [8]). We will also present some know results concerning the Fofana spaces (Lq,‘p)fi (see [4–7]. These spaces are defined as subspaces of Wiener Amalgam spaces W (Lq,‘p)(Rd), for 1 • p ˙ fi ˙ q • 1 (otherwise they are trivial). In particular we are able to describe them as dual Banach spaces and provide atomic characterizations of the predual.
Literature:
[1] R. R. Coifman and G. Weiss. Extensions of Hardy spaces and their use in analysis.
Bull. Amer. Math. Soc., 83(4):569–645, 1977.
[2] H. G. Feichtinger. On a new Segal algebra. Monatsh. Math., 92:269–289, 1981.
[3] H. G. Feichtinger. An elementary approach to Wiener’s third Tauberian theorem for
the Euclidean n-space. In Symposia Math., volume XXIX of Analisa Armonica, pages
267–301, Cortona, 1988.
[4] J. Feuto. Norm inequalities in generalized Morrey spaces. J. Fourier Anal. Appl.,
20(4):896–909, 2014.
[5] I. Fofana. Continuité de l’intégrale fractionnaire et espace (Lq,lp)fi (Continuity of the
fractional integral and (Lq,lp)fi). C. R. Acad. Sci., Paris, Sér. I, 308(18):525–527, 1989.
[6] I. Fofana and M. Sanogo. Fourier transform and compactness in (Lq,lp)fi and Mp,fi
spaces. Commun. Math. Anal., 11(2):139–153, 2011.
[7] B. A. Kpata, I. Fofana, and K. Koua. Necessary condition for measures which are
(lq,lp) multipliers. Ann. Math. Blaise Pascal, 16(2):339–353, 2009.
[8] N. Wiener. The Fourier Integral and certain of its Applications. Cambridge University
Press, Cambridge, 1933.