Convolution operators on discrete Hardy spaces (Q2729615)

The authors give some connections between the boundedness of convolution operators on the Hardy spaces \(H^p(\mathbb R^n)\) and some related operators on the discrete Hardy spaces \(H^p(\mathbb Z^n)\). NEWLINE\[NEWLINEH^p(\mathbb Z^n)=\{a\in \ell^p(\mathbb Z^n); \sup_{t>0}|P_t^d\ast a|\in \ell^p(\mathbb Z^n)\}NEWLINE\]NEWLINE with \(H^p\)-norm NEWLINE\[NEWLINE\|a\|_{H_{\max}}^p(\mathbb Z^n)=\|a\|_{\ell^p}+ \|\sup_{t>0}|P_t^d\ast a|\|_{\ell^p},NEWLINE\]NEWLINE where NEWLINE\[NEWLINEP_t^d(j)=C_nt/(t^2+j^2)^{(n+1)/2},\;n\neq 0,\;P_t^d(0)=0.NEWLINE\]NEWLINE This definition is equivalent to the one introduced by Coifman and Weiss. Let \(E_R\) be the set of slowly increasing \(C^\infty\) functions \(f\) with \(\text{supp}\widehat f\subset [-R, R]^n\). The main results of the authors are the following: Let \(0<p\leq 1\). Let \(m(\xi)\in L^\infty(\mathbb R^n)\) and \(K\) be the distribution defined by \((K\ast f)\widehat(\xi)=m(\xi)\widehat f(\xi)\).NEWLINENEWLINENEWLINE(1) Let \(\widehat\varphi\) be a Fourier multiplier on \(H^p(\mathbb R^n)\) so that \(\varphi \in E_R\) with \(0<R<1/2\). Then, \(\|K\ast f\|_{H^p(\mathbb R^n)}\leq \|f\|_{H^p(\mathbb R^n)}\) \((f\in H^p(\mathbb R^n)\cap L^2(\mathbb R^n))\) implies \(\|K^\varphi \ast a\|_{H^p(\mathbb Z^n)}\leq C\|a\|_{H^p(\mathbb Z^n)}\) \((a\in H^p(\mathbb Z^n)\), where \(K^\varphi (n)=(K\ast \varphi)(n)=(m\widehat\varphi)^\vee(n)\).NEWLINENEWLINENEWLINE(2) Let \(\widehat\varphi\in L^\infty(\mathbb R^n) \) satisfying \(\text{supp} \widehat\varphi \subset [-R, R]^n\), \(0<R<1/2\), and for some \(\varepsilon >0\) there exists \(h\in C_0^\infty(-\varepsilon , \varepsilon)^n\), \(h\equiv 1\) on \((-\varepsilon/2 , \varepsilon/2)^n\) so that \(h/\widehat\varphi \) is a Fourier multiplier on \(H^p(\mathbb R^n)\). Then, \(\|K_t^\varphi \ast a\|_{H^p(\mathbb Z^n)}\leq \|a\|_{H^p(\mathbb Z^n)}\) \((a\in H^p(\mathbb Z^n)\) implies \(\|K\ast f\|_{H^p(\mathbb R^n)}\leq C\|f\|_{H^p(\mathbb R^n)}\) \((f\in H^p(\mathbb R^n)\cap L^2(\mathbb R^n))\), where \(\widehat{K_t}(\xi)=\widehat K(t\xi)\). NEWLINENEWLINENEWLINEThey also give the maximal versions of the above. These results extend the known ones in the case \(1<p<\infty\) (de Leeuw, Kenig and Tomas, Auscher and Carro).