Embedding theorems for Müntz spaces (Q424837)

The paper deals with embeddings of Müntz spaces of \(L^1([0,1])\) in \(L^1(\mu)\). Let us recall that a Müntz subspace \(M_\Lambda^1\) of \(L^1([0,1])\) is the closure (relative to the \(L^1\)-norm) of the subspace spanned by the monomials \(x^{\lambda_n}\), where \(\Lambda=\{\lambda_n\}\) denotes an increasing sequence of positive real numbers. Moreover, it is well known that \(\sum1/\lambda_n<+\infty\) if and only if \(M_\Lambda^1\subsetneq L^1([0,1])\).NEWLINENEWLINEIn the following, this latter condition is always fulfilled. The functions belonging to \(M_\Lambda^1\) are actually real analytic (and extend to analytic function on the unit disc when the \(\lambda_n\)'s are integers). The geometry of Müntz spaces is still rather unknown (see the monograph of \textit{V. I. Gurariy} and \textit{W. Lusky} [Geometry of Müntz spaces and related questions. Lecture Notes in Mathematics 1870. Berlin: Springer (2005; Zbl 1094.46003)] for miscellaneous results on the subject).NEWLINENEWLINEOn the other hand, a large literature exists on embeddings of (more or less) classical spaces of analytic spaces (on the unit disc \({\mathbb D}\) for instance) in \(L^p({\mathbb D},\mu)\) spaces. Spaces like Hardy, Bergman, Dirichlet, \(\dots\) are maybe the most often considered. Very often, conditions involve Carleson measures.NEWLINENEWLINEHence, it is natural to wonder under which condition(s) we can embed \(M_\Lambda^1\) in some \(L^1(\mu)\). This work was initiated in the PhD thesis of \textit{Ihab Al Alam} (see [J. Math. Anal. Appl. 358, No. 2, 273--280 (2009; Zbl 1170.47011)], too). In the paper under review, some necessary and some sufficient conditions are given to ensure the boundedness or compactness of such embeddings. The authors consider the maximal function NEWLINE\[NEWLINEK(u)=\inf\Big\{C\geq0\,\Big|\;\|f\|_{\infty,[0,u]}\leq C\int_{[u,1]}|f(x)|dx\, ,\, \forall\, f\in M_\Lambda^1 \Big\} NEWLINE\]NEWLINE Among miscellaneous interesting results, let us mention the following sufficient criterion.NEWLINENEWLINETheorem 2.6. If \(K\in L^1(\mu)\), the embedding \(i_\mu:M^1_\Lambda\longrightarrow L^1(\mu) \) is bounded with norm less than \(\|K\|_{L^1(\mu)}\).NEWLINENEWLINETheorem 3.5 expresses the essential norm of \(i_\mu\).