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.
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Rubshtein, BZ.A., Grabarnik, G.Y., Muratov, M.A., Pashkova, Y.S. (2016). Embedding \(\varLambda _{\widetilde{V }}^{0}\) ⊆ X ⊆ MV* . In: Foundations of Symmetric Spaces of Measurable Functions. Developments in Mathematics, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-42758-4_12
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DOI: https://doi.org/10.1007/978-3-319-42758-4_12
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