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We prove the almost everywhere convergence of the inverse spherical transform of$L$_{$p$}bi-$K$-invariant functions on the group$SL$(2,$R$), 4/3<$p$≤2. The result appears to be sharp.

No abstract uploaded!

A two-dimensional periodic Schr\"{o}dingier operator is associated with every Lagrangian torus in the complex projective plane ${\mathbb C}P^2$. Using this operator we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional $W^{-}$ introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel--Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e. preserves the value of the energy functional. In particular, the deformations generated by Novikov--Veselov equations preserve the area of minimal Lagrangian tori.

Motivated by the Mario-Vafa formula of Hodge integrals and physicists' predictions on local Gromov-Witten invariants of toric Fano surfaces in a Calabi-Yau threefold, the third author conjectured a formula of certain Hodge integrals in terms of certain Chern-Simons invariants of the Hopf link. We prove this formula by virtual localization on moduli spaces of relative stable morphisms.