Study of harmonic analysis and applications to partial differential equations

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Arai studied harmonic analysis on negatively curved manifolds. Let M be a complete, simply connected Riemannian manifold whose sectional curvatures K_M satisfy -∞ < -k^2_2 【less than or equal】 K_M 【less than or equal】 -k^2_1 < 0, where k_1 and k_2 are positive constants. Arai obtained several results on elliptic harmonic functions on M. In particular he established fundamental part of harmonic analysis on M by proving theorems related to Hardy spaces, BMO, VMO, Carleson measures, Green's potential. As applications, he also studied Bloch function theory on manifolds and the regularity problem of degenerate harmonic measures. Ozawa studied by using real variable method nonlinear Schrodinger equations, nonlinear wave equations and nonlinear Krein-Gordon equations. Yajima obtained some results on the fundamental solutions of Schrodinger equations. Kanjin proved Paley's inequality for Jacobi expansions and studied the Hausdorff operator acting on real Hardy spaces.