In this paper, we prove endpoint estimates of L(log L) type for com-mutators of Littlewood-Paley operators, and the weighted weak type estimates for the commutators are also obtained.

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The paper deals with the characterization of lower (p, q)-type with respect to a proximate order of an entire function in terms of coefficients and exponents of its power series expansion. In particular, the problem proposed by Bajpai et al. [Trans. Amer. Math. Sac. (1975)], and subsequently, left open by Juneja et al. also [J. Reine Angew. Math. (1977)] has been solved.

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The superanalysis stands for doing elementary and real analysis on function spaces over the superspace _??_.^m/n with value R or C. Here, R and C are oo-dimensional Frechet-Grassmann algebras which play the role of R and C in the standard theory, respectively. Using this analysis, we construct a parametrix of the Pauli equation (=the Schrodinger equations with spin) on R³ from ‘classical objects’. More precisely, by using the differential operator representations of the Clifford alge-bra on the Grassmann algebra, we define the symbol of the Pauli equation as a super Hamiltonian function on the superspace. Solving the Hamilton-Jacobi and continuity equations corresponding to that Hamiltonian function, we construct a certain Fourier Integral Operator on superspace, which gives a parametrix of the Pauli equation. This parametrix is called “good” because it has not only the ordinary approximation prop-erties but also has the explicit dependence on the Planck constant h. The Lie product formula for these parametrices yields a desired evolutional operator of the Pauli equa-tion in the L²-scheme. In other words, we propose a quantization procedure of Feynman type for “classical mechanics with spin” using superanalysis.

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Let p be an odd prime and let a be an integer copirme to p. Denote by N(a, p) the number of pairs of integers b, c with bc_??_a (mod p), 1≤b, c<p and with b, c having different parity. The main purpose of this paper is to deduce an asymptotic formula for _??_.

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We shall characterize the quasi-Abelian varieties by some standard forms of period matrices. As applications we show fibration theorems for quasi-Abelian varieties.

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