Embedding theorem on spaces of homogeneous type
This paper deals with embedding theorems for Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. The main result shows that if \(-\varepsilon<s_1<s_2<\varepsilon\), then 1) \(\dot B_{p_2}^{s_2,q}\subset \dot B_{p_1}^{s_1,q}\) continuously for \(0<q\leq\infty\), \(\max\{{1\over 1+\varepsilon}, {1\over 1+\varepsilon+s_1}\}<p_1\leq\infty\), \(\max\{{1\over 1+\varepsilon}, {1\over 1+\varepsilon+s_2}\}<p_2\leq\infty\), \(y-\varepsilon<s_2-1/p_2=s_1-1/p_1<\varepsilon\). 2) \(\dot F_{p_2}^{s_2,q_2}\subset \dot F_{p_1}^{s_1,q_1}\) continuously for \(\max\{{1\over 1+\varepsilon}, {1\over 1+\varepsilon+s_1}\}<p_1, q_1<\infty\), \(\max\{{1\over 1+\varepsilon}, {1\over 1+\varepsilon+s_2}\} <p_2, q_2<\infty\), \(y-\varepsilon<s_2-1/p_2=s_1-1/p_1<\varepsilon\). To prove this result they need a discrete Calderón type reproducing formula on spaces of homogeneous type.
scientific article; zbMATH DE number 1890483