We introduce a new notion of “matrix potential” to nonlinear optical systems. In terms of a matrix potential g, we present a gauge field theoretic formulation of the Maxwell-Bloch equation that provides a semiclassical description of the propagation of optical pulses through resonant multi-level media. We show that the Bloch part of the equation can solved identically through g and the remainin...

Let g and φ be holomorphic maps on D such that φ(D) ⊂ D. Define Volterra composition operators Jg,φ and Ig,φ induced by g and φ as Jg,φf(z) = Z z 0 (f ◦ φ) (ζ) (g ◦ φ)′ (ζ) dζ and Ig,φf(z) = Z z 0 (f ◦ φ)′ (ζ) (g ◦ φ) (ζ) dζ for z ∈ D and f ∈ H(D), the space of holomorphic functions on D. In this paper, we characterize boundedness and compactness of these operators acting between weighted Bergm...

Suppose / is a holomorphic function on the open unit ball Bn of Cn. For 1 < p < oo and to > 0 an integer, we show that / is in Lp(Bn,dV) (with dV the volume measure) iff all the functions dmf/dza (\a\ = to) are in Lp(Bn,dV). We also prove that / is in the Bloch space of Bn iff all the functions dmf/dza (\a\ = m) are bounded on Bn. The corresponding result for the little Bloch space of Bn is est...