The continuous and discrete Bessel wavelet transformations were introduced by Pathak and Dixit [1], and analyzed their properties through the Hankel convolution approach developed by Haimo [2] and Hirschman [3]. However, many researchers have applied the Hankel transformation of type ( 1.75 ), and the Hankel convolution ( 1.79 ), to various function spaces, leading to numerous significant results. Liflyland and Moricz [4] considered the Hausdorff operator on the real Hardy space \(H^1(\mathbb {R})\) and proved the boundedness property of this operator by exploiting the Fourier and Hilbert transformations.