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Amer. Math. Soc. <strong>368</strong> (2016), 7119-7152 <span class="pull-right"><a href="https://www.copyright.com/openurl.do?issn=1088-6850&WT.mc.it=American%20Mathematical%20Society" target="_blank">Request permission</a></span> </dd> </dl> </header> <section id="Abstract"> <h2>Abstract:</h2> In this paper, we study the duality theory of the multi-parameter Triebel-Lizorkin spaces $\dot F^{\alpha ,q}_{p}(\mathbb R^{m})$ associated with the composition of two singular integral operators on $\mathbb {R}^m$ of different homogeneities. Such composition of two singular operators was considered by Phong and Stein in 1982. For $1<p<\infty$, we establish the dual spaces of such spaces as $(\dot F^{\alpha ,q}_{p}(\mathbb R^{m}))^*=\dot F^{-\alpha ,q’}_{p’}(\mathbb R^{m})$, and for $0<p\leq 1$ we prove $(\dot F^{\alpha ,q}_{p}(\mathbb R^{m}))^*=CMO^{-\alpha ,q’}_{p}(\mathbb {R}^m)$. We then prove the boundedness of the composition of two Calderón-Zygmund singular integral operators with different homogeneities on the spaces $CMO^{-\alpha ,q’}_{p}$. Surprisingly, such dual spaces are substantially different from those for the classical one-parameter Triebel-Lizorkin spaces $\dot {\mathcal {F}}^{\alpha ,q}_p(\mathbb R^{m})$. Our work requires more complicated analysis associated with the underlying geometry generated by the multi-parameter structures of the composition of two singular integral operators with different homogeneities. Therefore, it is more difficult to deal with than the duality result of the Triebel-Lizorkin spaces in the one-paramter settings. We note that for $0<p\leq 1$, $q=2$ and $\alpha =0$, $\dot F^{\alpha ,q}_{p}(\mathbb R^{m})$ is the Hardy space associated with the composition of two singular operators considered in Rev. Mat. Iberoam. <strong>29</strong> (2013), 1127–1157. 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Soc., Providence, R.I., 1979, pp. 85–94. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=545241">MR <strong>545241</strong></a></li> </ul> </details> </section> <section id="similar"> <details> <summary>Similar Articles</summary> <ul> <li> Retrieve articles in <em>Transactions of the American Mathematical Society</em> with MSC (2010): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B35&jrnl=one&onejrnl=tran">42B35</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B30&jrnl=one&onejrnl=tran">42B30</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B25&jrnl=one&onejrnl=tran">42B25</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B20&jrnl=one&onejrnl=tran">42B20</a> </li> <li> Retrieve articles in all journals with MSC (2010): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B35">42B35</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B30">42B30</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B25">42B25</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=42B20">42B20</a> </ul> </details> </section> <section id="additionalinformation"> <details> <summary>Additional Information</summary> <ul> <li><strong>Wei Ding</strong></li><li>Affiliation: School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing Normal University, Beijing, 100875, People’s Republic of China – and – School of Sciences, Nantong University, Nantong 226007, People’s Republic of China</li> <li>Email: dingwei@ntu.edu.cn</li> <li><strong>Guozhen Lu</strong></li><li>Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Repubic of China – and – Department of Mathematics, Wayne State University, Detroit, Michigan 48202</li> <li>MR Author ID: <a href='https://mathscinet.ams.org/mathscinet/search/authors.html?mrauthid=322112'>322112</a></li><li>Email: gzlu@wayne.edu</li> <li>Received by editor(s): February 14, 2014</li> <li>Received by editor(s) in revised form: September 5, 2014</li> <li>Published electronically: January 21, 2016</li> <li>Additional Notes: The research of the first author was partly supported by NNSF of China grants (No. 11371056 and No. 11271209) and the second author was partly supported by US NSF grant DMS#1301595. The second author is the corresponding author.</li> <li>© Copyright 2016 American Mathematical Society</li> <li>Journal: Trans. Amer. Math. Soc. <strong>368</strong> (2016), 7119-7152 </li> <li>MSC (2010): Primary 42B35, 42B30, 42B25, 42B20 <li>DOI: https://doi.org/10.1090/tran/6576</li> <li>MathSciNet review: <a href = "https://mathscinet.ams.org/mathscinet-getitem?mr=3471087">3471087</a></li> </ul> </details> </section> </article> </section> </article> </section> </main> </div>   <hr class="mx-auto"> <footer id="pagefooter" class="text-center"> <div class="container text-center"> <div class="row">  <div class="col-12 mb-1"> <p> American Mathematical Society · <a href="https://goo.gl/maps/ZvJdkQEC79p" title="Google map 201 Charles Street Providence, Rhode Island" aria-label="Google map to AMS headquarters"> 201 Charles Street Providence, Rhode Island 02904-2213 </a> · <a href="/reach" aria-label="Contact Us">Contact Us</a> </p> </div>  <script src="https://kit.fontawesome.com/2ad6d7fbd6.js" crossorigin="anonymous" defer></script>  <nav class="col-12 mb-1" aria-label="Social media links"> <ul class="list-inline"> <li class="list-inline-item"> <a href="https://www.facebook.com/amermathsoc" class="link-dark" title="Follow us on Facebook" aria-label="View the AMS on Facebook"> <i class="fa-brands fa-facebook-f" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://bsky.app/profile/amermathsoc.bsky.social" class="link-dark" title="Follow us on Bluesky" aria-label="View the AMS on Bluesky"> <i class="fa-brands fa-bluesky"></i></a></li> <li class="list-inline-item"> <a href="https://twitter.com/amermathsoc" class="link-dark" title="Follow us on X" aria-label="View the AMS on X"> <i class="fa-brands fa-x-twitter" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.linkedin.com/company/51001" class="link-dark" title="Follow us on LinkedIn" aria-label="View the AMS on LinkedIn"> <i class="fa-brands fa-linkedin-in" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.instagram.com/amermathsoc" class="link-dark" title="Follow us on Instagram" aria-label="View the AMS on Instagram"> <i class="fa-brands fa-instagram" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.youtube.com/amermathsoc" class="link-dark" title="Follow us on YouTube" aria-label="View the AMS on YouTube"> <i class="fa-brands fa-youtube" aria-hidden="true"></i></a></li> </ul> </nav>  <div class="col-12"> <p><small> © <a href="/copyright" aria-label="Copyright policy">Copyright</a> <span id="copyYear">2024</span> American Mathematical Society · <a href="/privacy" aria-label="Privacy statement">View Our Privacy Statement</a> · <a href="/tou" aria-label="Terms of use">Terms of Use</a> · <a href="/accessibility" aria-label="Accessibility policy">Accessibility and AMS Online Content</a> </small></p> <p><small> AMS, American Mathematical Society, the tri-colored AMS logo, and Advancing research, Creating connections, are trademarks and service marks of the American Mathematical Society and registered in the U.S. Patent and Trademark Office. </small></p> </div> </div> </div> <script> // Dynamically update the copyright year document.getElementById('copyYear').textContent = new Date().getFullYear(); </script> </footer>