The Fourier transforms of Lipschitz functions on certain domains (Q1380766)

The Fourier transforms (or coefficient in the periodic case) of various Lipschitz functions are studied, with special emphasis on radial functions. For \(\text{Lip}(\alpha_1, \alpha_2,p)\), where \(1\leq p<\infty\) and \(0<\alpha_1,\alpha_2\leq 1\), the following two definitions are used; either \[ \| f(x_1+ h_1, x_2+ h_2)- f(x_1, x_2+ h_2)- f(x_1+ h_1,x_2)+ f(x_1, x_2)\|_p= O(h^{\alpha_1}_1 h^{\alpha_2}_2)\tag{i} \] or \[ \| f(x_1+ h_1, x_2+ h_2)- f(x_1, x_2)\|_p= O(h^{\alpha_1}_1+ h^{\alpha_2}_2).\tag{ii} \] A brief comparison between the results of the present paper and those obtained for the Hankel transform of certain Lipschitz functions is also included.