Equivalent norms, Hardy-Littlewood-type theorems, and their applications

The primary goal of this paper is to develop methods for investigating equivalent norms and Hardy-Littlewood-type theorems on Lipschitz-type spaces of analytic and complex-valued harmonic functions. First, we provide characterizations of equivalent norms on these spaces. Furthermore, we establish Hardy-Littlewood-type theorems for complex-valued harmonic functions. These results improve and extend the main findings of Dyakonov (1997) and Dyakonov (2005). Additionally, we apply the derived equivalent norms and Hardy-Littlewood-type theorems to the study of composition operators between Lipschitz-type spaces.