On the characterization of Hardy-Besov spaces on the dyadic group and its applications

C. Watari [12] obtained a simple characterization of Lipschitz classes ${\displaystyle Li{p}^{\left(p\right)}\alpha \left(W\right)(1\ge p\ge \infty ,\alpha >0)}$ on the dyadic group using the ${\displaystyle L^p}$-modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces ${\displaystyle B^{\alpha }\_\{p,q\}}$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces ${\displaystyle B^{\alpha }\_\{p,q\}}$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the type of the Maz'ya inequality, a weak type estimate for maximal Cesàro means and a sufficient condition of absolute convergence of Walsh-Fourier series.