Equivalence Between A Harmonic Form and A Closed Co-Closed Form in Both Lq and Non-Lq Spaces
For a differential k-form Ï‰ on a complete non-compact manifold, we establish an equivalent relation between a harmonic form and a closed co-closed form. We extend this equivalence from Ï‰ in L 2 spaces to Ï‰ with 2-balanced growth including L 2 spaces and non-L 2 spaces. Especially for a simple differential k-form Â¯Ï‰ on a complete non-compact manifold, we generalize this equivalence from Â¯Ï‰ in L q spaces to Â¯Ï‰ with 2-balanced growth including L q spaces and non-L q spaces for 2 â‰¤ q < 3. Our research findings recapture the work of Andreotti and Vesentini. Our ideas and calculation methods in this paper could provide a new way of broadening L q spaces to non-L q spaces in a variety of energy for differential forms.