For a fixed integer n ≥ 1, let p = 2nℓ + 1 be a prime number with an odd prime number ℓ and let F = Fp,ℓ be the real abelian field of conductor p and degree ℓ. When n ≤ 21, we show that a prime number r does not divide the class number hF of F whenever r is a primitive root modulo ℓ with the help of computer. This generalizes a result of Jakubec and Metsänkylä for the case n = 1.
In this paper we prove the boundedness of the generalized fractional integral operator Iρ on generalized Campanato spaces with variable growth condition, which is a generalization and improvement of previous results, and then, we establish the boundedness of Iρ on their bi-preduals. We also prove the boundedness of Iρ on their preduals by the duality.
In this paper, we prove that any non-constant real rational function appears as a time transformation of a caloric morphism, mapping which preserves caloric functions, between semi-eucledean spaces.