On double sine and cosine transforms,Lipschitz and Zygmund classes

We consider complex-valued functions f ∈ L ¹(R ₊²), where R ₊:= [0,∞), and prove sufficient conditions under which the double sine Fourier transform [^(f)]_sshat f_{ss} and the double cosine Fourier transform [^(f)]_cchat f_{cc} belong to one of the two-dimensional Lipschitz classes Lip(α, β) for some 0 < α, β ≤ 1; or to one of the Zygmund classes Zyg(α, β) for some 0 < α, β ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L ¹(R ₊²).