Ramsey-type theorems for sets satisfying a geometric regularity condition

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• 作者：P. Hagelstein ; ¹ ; ^{paul_hagelstein@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; D. Herden ^{daniel_herden@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; D. Young² ; ^{daniel_young1@baylor.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ramsey theory ; Regularity condition ; Maximal operators
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：951-956
• 全文大小：291 K

文摘

We consider Ramsey-type problems associated to collections of sets in ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=2d954dd1e588cb5ae7e7640c55e897ae" title="Click to view the MathML source">Rⁿ${\mathbb{R}}^n$ satisfying a standard geometric regularity condition. In particular, let ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=46ee81650a7453b0d925f4c039ba4f86"> be a collection of measurable sets in ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=2d954dd1e588cb5ae7e7640c55e897ae" title="Click to view the MathML source">Rⁿ${\mathbb{R}}^n$ such that every ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=c56f6f6095ac169966f7a004cb7d3e20" title="Click to view the MathML source">R_j$R_j$ is contained in a cube ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=2c9619c098ef77fbe193ed88030ec2c2" title="Click to view the MathML source">Q_j$Q_j$ whose sides are parallel to the axes and such that ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=b1752a799947940fdc01160c90812b10" title="Click to view the MathML source">|R_j|/|Q_j|≥ρ>0. Moreover, suppose that there exists ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=95712db7778f6a09483cae1ce281a773" title="Click to view the MathML source">0<γ<∞$0<\gamma <\infty$ such that ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=93ee0e433a34221d292360089d2d6201" title="Click to view the MathML source">|R_j|/|R_k|≤γ for every ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=ebb206b67a70e850759c100c069abfee" title="Click to view the MathML source">j,k$j,k$. We prove that there exists a subcollection of ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=46ee81650a7453b0d925f4c039ba4f86"> consisting of at least ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=a0dd968377980c227587eda730175783" title="Click to view the MathML source">R(N) sets that either have a point of common intersection or that are pairwise disjoint, where ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=1150b1c97d4d9893e8fad4e53e382d1f">. If the sets in the collection ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=34a501a5e5a180f05ba11025412b4e9a" title="Click to view the MathML source">{R_j} are convex, we obtain the improved Ramsey estimate ser=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=66f9d48073a0c53f5abf275cbd3e6a69" title="Click to view the MathML source">R(N)≥(3⁻ⁿρN)^1/2. Applications of these results to weak type bounds of geometric maximal operators are provided.