Harmonic Analysis for Some Orthogonal Expansions

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Our research results are summarized as follows. The head investigator Kanjin has obtained Paley's inequality and Hardy's inequality with respect to the Jacobi expansions. The classical Peley's inequality and Hardy's inequality are two of the most familiar inequalities on the Fourier coefficients of functions in the Hardy space of certain analytic functions in the unit disc. The inequalities were originally proved by complex method. It is difficult to study orthogonal expansions by using complex method. Recent development of the real Hardy space theory, especially the atomic decomposition characterization or the real Hardy space and BMO space duality, allows to discuss problems on inequalities with respect to orthogonal expansions. Our Hardy's inequality have proved by applying the atomic decomposition to the Jacobi function system and Our Paley's inequality has gotten by using the real Hardy space and BMO space duality. Further, he has studied the Cesaro operator and, generally, the Hausdorff operator. The result says that the Hausdorff operator is bounded on the real Hardy space with parameter p smaller than one under some conditions.The investigator Tsuchiya has investigated convergence of Dirichlet forms of diffusion process without assuming that the underlying measures are fixed or compatible with a fixed one. Ichinose has obtained more results on the self-adjoint Trotter-Kate product formula with operator norm. Sato has considered Marcinkiewicz integrals arising from rough kernels satisfying LlogL condition on the unit (n-l)-sphere and proved the weak type (1,1) estimates. Tohge has studied a Riccati differential equation whose coefficient is expressible in terms of a special Weierstrass pe-function and shown that all the solutions are meromorphic.