Embeddings of Müntz Spaces in $L^{∞ }(\unicode[STIX]{x1D707})$. Issue 1 (9th January 2019) - Al Alam, Ihab; Lefèvre, Pascal
Abstract: In this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{∞ }{\hookrightarrow}L^{∞ }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0, 1]$ and $M_{\unicode[STIX]{x1D6EC}}^{∞ }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1}, \unicode[STIX]{x1D707}_{2}}:L^{∞ }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{∞ }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$ .