On composition operators acting between Hardy and weighted Bergman spaces

We present a unified approach to some known and some new criteria for the boundedness and compactness of composition operators mapping a weighted Bergman space Aαp into another weighted Bergman space Aβq, where q⩾pq⩾p and α,β>-1α,β>-1, also obtaining some asymptotic formulas for the essential norm. The results are also valid in the limit cases when at least one of the spaces is a Hardy space (i.e., when αα or β=-1β=-1) and complement the existing results by various authors.