For a ≥β≥ -1/2 let △（x） = （2shx）^2α＋1 （2chx）2β＋1 denote the weight function on R＋ and L^1 （△） the space of integrable functions on R＋ with respect to △（x）dx, equipped with a convolution structure...For a ≥β≥ -1/2 let △（x） = （2shx）^2α＋1 （2chx）2β＋1 denote the weight function on R＋ and L^1 （△） the space of integrable functions on R＋ with respect to △（x）dx, equipped with a convolution structure. For a suitable Ф ∈ L^1 （△）, we put Фt（x） = t^-1 △（x）^-1 △（x/t）Ф（x/t） for t 〉 0 and define the radial maximal operator MФ, as usual manner. We introduce a real Hardy space H^1 （△） as the set of all locally integrable functions f on R＋ whose radial maximal function MФ （f） belongs to L^1 （△）. In this paper we obtain a relation between H^1 （△） and H^1 （R）. Indeed, we characterize H^1 （△） in terms of weighted H^1 Hardy spaces on R via the Abel transform of f. As applications of H^1 （△） and its characterization, we shall consider （H^1 （△）,L^1 （△））-boundedness of some operators associated to the Poisson kernel for Jacobi analysis： the Poisson maximal operator Me, the Littlewood-Paley g-function and the Lusin area function S. They are bounded on L^p（△） for p 〉 1, but not true for p = 1. Instead, Mp, g and a modified Sa,r are bounded from H^1 （△） to L^1 （△）.展开更多
Abstract. Let （R＋,＊,A） be the Jacobi hypergroup. We introduce analogues of the Littlewood-Paley g function and the Lusin area function for the Jacobi hypergroup and consider their （H^1, L^1 ） boundedness. Althou...Abstract. Let （R＋,＊,A） be the Jacobi hypergroup. We introduce analogues of the Littlewood-Paley g function and the Lusin area function for the Jacobi hypergroup and consider their （H^1, L^1 ） boundedness. Although the g operator for （R＋,＊,A） possesses better property than the classical g operator, the Lusin area operator has an obstacle arisen from a second convolution. Hence, in order to obtain the （H^1,L^1） estimate for the Lusin area operator, a slight modification in its form is required.展开更多
The classical Hardy theorem asserts that f and its Fourier transform f can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU（1, ...The classical Hardy theorem asserts that f and its Fourier transform f can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU（1, 1） there are infinitely many “good” functions in the sense that f and its spherical Fourier transform y both have good decay. In this paper, we shall characterize such functions on SU（1, 1）.展开更多
基金Partly supported by Grant-in-Aid for Scientific Research (C), No. 20540188, Japan Society for the Promotion of Science
文摘For a ≥β≥ -1/2 let △（x） = （2shx）^2α＋1 （2chx）2β＋1 denote the weight function on R＋ and L^1 （△） the space of integrable functions on R＋ with respect to △（x）dx, equipped with a convolution structure. For a suitable Ф ∈ L^1 （△）, we put Фt（x） = t^-1 △（x）^-1 △（x/t）Ф（x/t） for t 〉 0 and define the radial maximal operator MФ, as usual manner. We introduce a real Hardy space H^1 （△） as the set of all locally integrable functions f on R＋ whose radial maximal function MФ （f） belongs to L^1 （△）. In this paper we obtain a relation between H^1 （△） and H^1 （R）. Indeed, we characterize H^1 （△） in terms of weighted H^1 Hardy spaces on R via the Abel transform of f. As applications of H^1 （△） and its characterization, we shall consider （H^1 （△）,L^1 （△））-boundedness of some operators associated to the Poisson kernel for Jacobi analysis： the Poisson maximal operator Me, the Littlewood-Paley g-function and the Lusin area function S. They are bounded on L^p（△） for p 〉 1, but not true for p = 1. Instead, Mp, g and a modified Sa,r are bounded from H^1 （△） to L^1 （△）.
基金partly supported by Grant-in-Aid for Scientific Research (C) No.24540191, Japan Society for the Promotion of Science
文摘Abstract. Let （R＋,＊,A） be the Jacobi hypergroup. We introduce analogues of the Littlewood-Paley g function and the Lusin area function for the Jacobi hypergroup and consider their （H^1, L^1 ） boundedness. Although the g operator for （R＋,＊,A） possesses better property than the classical g operator, the Lusin area operator has an obstacle arisen from a second convolution. Hence, in order to obtain the （H^1,L^1） estimate for the Lusin area operator, a slight modification in its form is required.
基金Project supported by Grant-in-Aid for Scientific Research(C)of Japan(No.16540168)the National Natural Science Foundation of China(No.10371004).
文摘The classical Hardy theorem asserts that f and its Fourier transform f can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU（1, 1） there are infinitely many “good” functions in the sense that f and its spherical Fourier transform y both have good decay. In this paper, we shall characterize such functions on SU（1, 1）.
