Besov space is an effective tool for measuring regular properties of functions and is a generalization of Sobolev space. Exploiting the results of Fourier analysis, the characterizations of the spaces $$B_{p,q}^s(\mathbb {R}^n)$$ and $$F_{p,q}^s(\mathbb {R}^n)$$ were introduced by Triebel [1]. Motivated from the result of multiplier theorem of Triebel [1], boundedness results of the wavelet transform in Besov space $$B_{p,q}^s(\mathbb {R}^n)$$ and Triebel-Lizorkin space $$F_{p,q}^s(\mathbb {R}^n)$$ for $$s\in \mathbb {R}$$ and $$0