We consider complex-valued functions $f \in L^1(\mathbf{R}^2_+)$, where $\mathbf{R}_+ := [0,\infty)$, and prove sufficient conditions under which the double sine Fourier transform $\hat{f}_{ss}$ and the double cosine Fourier transform $\hat{f}_{cc}$ belong to one of the two-dimensional Lipschitz classes $Lip(\alpha,\beta )$ for some $0 < \alpha,\beta \leq 1$; or to one of the Zygmund classes Zyg$(\alpha,\beta )$ for some $0 < \alpha,\beta \leq 2$. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions $f \in L^1(\mathbf{R}^2_+)$.

}, issn = {1573-8175}, doi = {https://doi.org/10.1007/s10496-011-0351-9}, url = {http://global-sci.org/intro/article_detail/ata/4607.html} }