Besov-type spaces of variable smoothness on rough domains

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• 作者：A.I. Tyulenev ^{tyulenev-math@yandex.ru" class="auth_mail" title="E-mail the corresponding author tyulenev@mi.ras.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：46E35
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：145
• 期：Complete
• 页码：176-198
• 全文大小：830 K

文摘

The paper puts forward new Besov spaces of variable smoothness 46X16301699&_mathId=si1.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=f9ada8c4754beec6d49edddfb00c54ac">46X16301699-si1.gif">$B_{p,q}^{{\phi }_0}(G,\left\{t_k\right\})$ and 46X16301699&_mathId=si2.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=bffa9a2e9b60d76d1df7a645dea880d9">46X16301699-si2.gif">${\tilde{B}}_{p,q,r}^l(\Omega ,\left\{t_k\right\})$ on rough domains. A domain 46X16301699&_mathId=si3.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=cca838d11aed1a4971df776ce63dd9be" title="Click to view the MathML source">G$G$ is either a bounded Lipschitz domain in 46X16301699&_mathId=si4.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=4b255e55fed7e72b6e1ea2b722fdb8f3" title="Click to view the MathML source">Rⁿ${\mathbb{R}}^n$ or the epigraph of a Lipschitz function, a domain 46X16301699&_mathId=si5.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=1fbacf1f8964d2bc90fc5f6a6e384fa5">46X16301699-si5.gif">$\Omega$ is an 46X16301699&_mathId=si6.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=80839f36a035b8546b40bc3c59022a3a" title="Click to view the MathML source">(ε,δ)$(\epsilon ,\delta )$-domain. These spaces are shown to be the traces of the spaces 46X16301699&_mathId=si7.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=556aa3567b2753159ec6db367756d60a">46X16301699-si7.gif">$B_{p,q}^{{\phi }_0}({\mathbb{R}}^n,\left\{t_k\right\})$ and 46X16301699&_mathId=si8.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=b51e12356dd8e271c82393f79bfaecff">46X16301699-si8.gif">${\tilde{B}}_{p,q,r}^l({\mathbb{R}}^n,\left\{t_k\right\})$ on domains 46X16301699&_mathId=si3.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=cca838d11aed1a4971df776ce63dd9be" title="Click to view the MathML source">G$G$ and 46X16301699&_mathId=si5.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=1fbacf1f8964d2bc90fc5f6a6e384fa5">46X16301699-si5.gif">$\Omega$, respectively. The extension operator 46X16301699&_mathId=si11.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=c28ab889cfbc83b09833a1e7e45c069c">46X16301699-si11.gif">${Ext}_1:B_{p,q}^{{\phi }_0}(G,\left\{t_k\right\})\to B_{p,q}^{{\phi }_0}({\mathbb{R}}^n,\left\{t_k\right\})$ is linear, the operator 46X16301699&_mathId=si12.gif&_user=111111111&_pii=S0362546X16301699&_rdoc=1&_issn=0362546X&md5=ab5604a654675964d9bf227a4392a011">46X16301699-si12.gif">${Ext}_2:{\tilde{B}}_{p,q,r}^l(\Omega ,\left\{t_k\right\})\to {\tilde{B}}_{p,q,r}^l({\mathbb{R}}^n,\left\{t_k\right\})$ is nonlinear. As a corollary, an exact description of the traces of 2-microlocal Besov-type spaces and weighted Besov-type spaces on rough domains is obtained.