A critical topology for L^p-Carleman classes with 0

Abstract

In this paper, we explore a sharp phase transition phenomenon which occurs for L^p-Carleman classes with exponents 0<p<1. These classes are defined as for the standard Carleman classes, only the L^infty -bounds are replaced by corresponding L^p-bounds. We study the quasinorms |u|p,M=supn≥0||u(n)||pMn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left||u\\right||_{p,\\mathcal {M}}=\\sup _{n\\ge 0}\\frac{||u^{(n)}||_p}{M_n}, \\end{aligned}$$\\end{document}for some weight sequence mathcal {M}={M_n}_n of positive real numbers, and consider as the corresponding L^p-Carleman space the completion of a given collection of smooth test functions. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the L^p-Carleman class. A particular degenerate instance is when M_n=1 for 0le nle k and M_n=+infty for n>k. This would give the L^p-Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these L^p-Sobolev spaces are highly degenerate for 0<p<1. Indeed, the canonical map W^{k,p}rightarrow L^p fails to be injective, and there is even an isomorphism Wk,p≅Lp⊕Lp⊕⋯⊕Lp,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} W^{k,p}\\cong L^p\\oplus L^p\\oplus \\cdots \\oplus L^p, \\end{aligned}$$\\end{document}corresponding to the canonical map fmapsto (f,f',ldots ,f^{(k)}) acting on the test functions. This means that e.g. the function and its derivative lose contact with each other (they “disconnect”). Here, we analyze this degeneracy for the more general L^p-Carleman classes defined by a weight sequence mathcal {M}. If mathcal {M} has some regularity properties, and if the given collection of test functions is what we call (p,theta )-tame, then we find that there is a sharp boundary, defined in terms of the weight mathcal {M}: on the one side, we get Douady–Peetre’s phenomenon of “disconnexion”, while on the other, the completion of the test functions consists of C^infty -smooth functions and the canonical map fmapsto (f,f',f'',ldots ) is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the L^p setting, with 0<p<1.