Research on Fourier integrals of several variables

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(1)We proved the weighted weak type (1,1) estimates for the Calderon-Zygmund type singular integrals. These operators are defined by certain rough kernels. We assume that the kernel satisfies a certain size condition and a cancellation condition along with a Dini type condition. These results are a generalization to R^n, n【greater than or equal】3, of the results of A. Vargas on the weak (1,1) estimates for the singular integrals with homogeneous kernels. Also, they are a generalization to the case of inhomogeneous kernels on R^n, n【greater than or equal】2, of the results of A. Vargas. The weighted weak type estimates for the Littlewood-Paley functions are also shown by assuming analogous conditions for the kernels.(2)For certain classes of pseudo-differential operators, we proved L^2_ω-L^2_ω, L^1_ω-L^<1,∞>_ω and H^1_ω-L^1_ω estimates. We proved L^2_ω-L^2_ωestimates for a pseudo-differential operators with a symbol satisfying a minimal regularity condition, where the weight ω is in A_1 … More of Muckenhoupt weight class. This improves a result of K. Yabuta. The L^1_ω-L^<1,∞>_ω and H^1_ω-L^1_ω estimates were proved by applying Carbery's method.(3)We studied the singular integrals associated with the variable surfaces of revolution. We treated the rough kernel case where the singular integral is defined by an H^1 kernel function on the sphere S^<n-1>. We proved the L^p boundedness of the singular integral for 1<p【less than or equal】2 assuming that a certain lower dimensional maximal operator is bounded on L^s for all s>1. We also studied the (L^p, L^r) boundedness for the fractional integrals associated with the surfaces of revolution.(4)We proved some weighted estimates for the maximal functions associated with certain Fourier multipliers of Bochner-Riesz type.(5)We considered certain non-regular pseudo-differential operators T_σ and studied the question of their boundedness on the weighted Triebel-Lizorkin and Besov spaces. In particular, we substantially relaxed the regularity condition on the symbol σ due to Bourdaud for T_σ to be bounded on the Sobolev spaces H^s_p (p【greater than or equal】2). Less