Celebratio Mathematica — Zygmund

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A. Zygmund: “On Fourier coefficients and transforms of functions of two variables,” Studia Math. 50 (1974), pp. 189–201. To Marceli Stark. MR 0387950 Zbl 0278.42005 article

Let \( f(x_1,x_2) \) be a function of two variables, of period 1 in each, and let \( c_{\mu} = c_{m,n} \) be the Fourier coefficients of \( f \). Then, if \[ 1 \leq p \leq \tfrac{4}{3} \quad\text{and}\quad q = \tfrac{1}{2}p^{\prime} = \tfrac{1}{2}p/(p-1) ,\] we have \[ \Bigl\{ \sum_{|\mu| = r}|c_{\mu}|^q \Bigr\}^{1/q} \leq A_p\|f\|_p \qquad(A_p = 5^{1/p^{\prime}}) \] for all \( r > 0 \). There is a corresponding result for Fourier transforms of functions \( f\in L^p(\mathbf{R}^2) \), \( 1 \leq p < \frac{4}{3} \), but the previous \( q = \frac{1}{2}p^{\prime} \) has to be replaced by \( q = \frac{1}{3}p^{\prime} \). Moreover, the result fails in the extreme case \( p = \frac{4}{3} \). The results are strictly two-dimensional.