| Sumario: | In this work, we shall provide an F (R) gravity theoretical framework for solving the H0-tension. Specif- ically, by exploiting the F (R) gravity correspondence with a scalar-tensor theory, we shall provide a condition in which when it is satisfied, the H0-tension is alleviated. The condition that remedies the H0- tension restricts the corresponding F (R) gravity, and we present in brief the theoretical features of the constrained F (R) gravity theory in both the Jordan and Einstein frames. The condition that may remedy the H0-tension is based on the existence of a metastable de Sitter point that occurs for redshifts near the recombination. This metastable de Sitter vacuum restricts the functional form of the F (R) gravity in the Jordan frame. We also show that by appropriately choosing the F (R) gravity, along with the theoretical solution offered for the H0-tension problem, one may also provide a unified description of the inflationary era with the late-time accelerating era, in terms of two extra de Sitter vacua. We propose a new approach to F (R) gravity by introducing a new class of integral F (R) gravity functions, which may be wider than the usual class expressed in terms of elementary F (R) gravity functions. Finally, the Einstein frame inflationary dynamics formalism is briefly discussed.
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