Fourier coefficients of functions in \(H^p(S)\) (Q2744446)

Let \(H^p(S)\) be the Hardy space on the unit spherical surface \(S\) in the \(n\)-dimensional complex space \(\mathbb{C}^n\). For an arbitrary function \(f\in H^p(S)\) the Fourier coefficients are defined by NEWLINE\[NEWLINEa_\alpha= {(|\alpha|+ n-1)!\over \alpha!(n- 1)!} \int_S f(\xi) \overline\xi^\alpha \sigma d\sigma(\xi),\quad\alpha\in Z^{+n},NEWLINE\]NEWLINE here \(Z^{+n}\) is the \(n\)-group of nonnegative integers and \(f(\xi)\sim\sum_{\alpha\in Z^{+n}} a_\alpha\xi^\alpha\) is the Fourier expansion.NEWLINENEWLINENEWLINEIn this article, several inequalities for the Fourier coefficients of functions of several complex variables of the space \(H^p\) are obtained. The methods of proof are different from the relevant situation for one variable (Hausdorff-Young inequality, Hardy-Littlewood inequality) to a large extent. The authors prove three theorems, for example:NEWLINENEWLINENEWLINETheorem 1: Let \(1< p\leq 2\), \({1\over p}+{1\over q}= 1\). If \(f\in H^p(S)\) and \(f(\xi)\sim \sum_{\alpha\in Z^{+n}} a_\alpha\xi^\alpha\), then NEWLINE\[NEWLINE\Biggl\{\sum_{\alpha\in Z^{+n}} \Biggl[{\alpha!(n- 1)!\over (|\alpha|+ n-1)!}\Biggr]^{q-1} |a_\alpha|^q\Biggr\}^{{1\over q}}\leq\|f\|_p;NEWLINE\]NEWLINE conversely, if NEWLINE\[NEWLINE\sum_{\alpha\in Z^{+n}} \Biggl[{\alpha!(n- 1)!\over (|\alpha|+ n-1)!}\Biggr]^{p-1} |a_\alpha|^p< \infty,NEWLINE\]NEWLINE then there is an \(f\in H^2(S)\), \(f(\xi)\sim \sum_{\alpha\in Z^{+n}} a_\alpha\xi^\alpha\), and NEWLINE\[NEWLINE\|f\|_q\leq C_q\Biggl\{\sum_{\alpha\in Z^{+n}} \Biggl[{\alpha!(n- 1)!\over (|\alpha|+ n-1)!}\Biggr]^{p-1}|a_\alpha|^p\Biggr\}^{{1\over p}},NEWLINE\]NEWLINE for some constant \(C_q\) depending only on \(q\).