Abstract
If $$\varphi $$ is an entire function, the superposition operator $$S_\varphi $$ is defined in the space $${\mathcal {H}}{\mathcal {o}}{\mathcal {l}}({\mathbb {D}})$$ of all analytic functions f in the unit disc $${\mathbb {D}}$$ , by $$S_{\varphi }(f)=\varphi \circ f$$ . We consider the mixed norm spaces of Hardy type $$H(p,q,\alpha )$$ ( $$0<p,q\le \infty $$ , $$\alpha >0$$ ). In this work we provide a complete characterization of those entire functions $$\varphi $$ so that the superposition operator $$S_\varphi $$ maps $$H(p,q,\alpha )$$ into $$H(s,t,\beta )$$ for any two triplets of admissible parameters $$(p,q,\alpha )$$ and $$(s,t,\beta )$$ . We also prove that superposition operators mapping a mixed norm space into another are bounded and continuous.
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