Embedding Λ V ̃ 0 $$\varLambda _{\widetilde{V }}^{0}$$ ⊆ X ⊆ MV*

Abstract

In this chapter, we prove the embedding theorem for classes of symmetric spaces having the same fundamental functions. The embedding theorem asserts that for every symmetric space X with a given fundamental function V = φ X , there are continuous embeddings \(\boldsymbol{\varLambda }_{\widetilde{V }}^{0} \subseteq \mathbf{X} \subseteq \mathbf{M}_{V _{{\ast}}}\). This means that the minimal part \(\boldsymbol{\varLambda }_{\widetilde{V }}^{0}\) of the Lorentz space \(\boldsymbol{\varLambda }_{\widetilde{V }}\) is the smallest symmetric space whose (concave) fundamental function \(\widetilde{V }\) is equivalent to V, while the Marcinkiewicz space \(\mathbf{M}_{V _{{\ast}}}\) is the largest symmetric space X with \(\varphi _{\mathbf{X}} =\varphi _{\mathbf{M}_{V_{{\ast}}}} = V\). The renorming theorem and other consequences of the embedding theorem are considered.