Harmonic Analysis on Orthogonal Expansions

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Our research results are summarized as follows. Head investigator Kanjin has obtained Hardy's inequalities with respect to the Hermite and Laguerre expansions. The classical Hardy's inequality is a well-known inequality on the Fourier coefficients of functions in the Hardy space of certain analytic functions in the unit disc. The inequality was originally proved by complex method. It is difficult to study the orthogonal polynomial expansions by using complex method. Recent development of the real Hardy space theory, especially the atomic decomposition characterization of the real Hardy space allows to discuss the problem on the inequalities with respect to the orthogonal expansions. Our inequalities have proved by applying the atomic decomposition to the Hermite and Laguerre systems. By our method we have also gotten Hardy's inequality for the Hankel transforms. Further, we have studied the discrete Hardy space and obtained the molecular characterization of the space. As its applications, we have proved the theorem of fractional integration and the Marcinkiewicz type multiplier theorem for the discrete Hardy space.Investigator Ichinose has proved the Lie-Trotter-Kato product formula with operator norm. Tohge has gotten a result on the relation between the classical Nevanlinna theory and linear differential equations. Tsuchiya has shown Feller property for a Markov process obtained by superposing two diffusion processes in two domains under Holder condition for coefficients. Sato has considered Al-weights and proved weighted weak type (1,1) estimates for oscillatory singular integrals with kernels satisfying a Dini condition.