Fine differentiability and the HardyLittlewood maximal function

Key Points

• To introduce a relaxed version of the Lipschitz number and analyze differentiability in fine topology.
• Introduced the relaxed Lipschitz number using 1-fine topology.
• Analyzed the conditions for the 1-fine differentiability of functions with bounded variation.
• Used Stepanoff-type theorem to establish almost everywhere differentiability results.
• Function $w$ is shown to be 1-finely differentiable almost everywhere where $ ext{Lip}_w^{fine}< ext{∞}$.
• Demonstrated that the Hardy-Littlewood maximal function is 1-finely differentiable almost everywhere.