2026-09-21T05:30:17Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/19178 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Ruan, Zhong-Jin Choi, Changsun 2011-05-07T11:59:20Z 2011-05-07T11:59:20Z 10000-01-01 1995 In Chapter 1 we sharpen Burkholder's inequality $\mu(\vert v\vert\geq1)\leq2\Vert u\Vert\sb1$ for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality $\mu(\vert v\vert\geq1)\leq K\Vert u\Vert\sb1$ under the assumptions that $\vert v(\xi)\vert\leq\vert u(\xi)\vert, \vert\nabla v\vert\leq\vert\nabla u\vert$ and the extra assumption that $\nabla u\cdot\nabla v$ = 0. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant 1 $<K<$ 2, found by Davis, is the best constant in Kolmogorov's weak-type inequality for conjugate functions. In Chapter 2 we get norm inequalities. Let ($\Omega,{\cal F},P$) be a probability space with filtration (${\cal F}\sb{n}).$ Let f be a nonnegative submartingale and g be an adapted sequence. Let d be the difference sequence of f and e of $g{:} f\sb{n}=\sum\limits\sbsp{k=0}{n}\ d\sb{k}$ and $g\sb{n}=\sum\limits\sbsp{k=0}{n}\ e\sb{k}, n\ge 0.$ We prove $\Vert g\Vert\sb{p}\le(r-1)\Vert f\Vert\sb{p}$ under the assumption that $\vert e\sb{n}\vert\le\vert d\sb{n}\vert$ for $n\ge 0$ and $\vert{\rm I\!E}(e\sb{n}\ \mid\ {\cal F}\sb{n-1})\vert\le\alpha\vert{\rm I\!E}(d\sb{n}\ \mid\ {\cal F}\sb{n-1})\vert$ for $n\ge 1.$ Here 0 $\le\alpha\le$ 1, 1 $<p<\infty$ are constants, $\Vert f\Vert\sb{p}={\rm sup}\Vert f\sb{n}\Vert\sb{p}$ and $r=\max\{(\alpha+1)p,p/(p-1)\} .$ We also get similar inequalities $\Vert v\Vert\sb{p}\le(r-1)\Vert u\Vert\sb{p}$ and $\Vert\vert Y\Vert\vert\sb{p}\le(r-1)\Vert\vert X\Vert\vert\sb{p}$ where u, v are smooth functions and X, Y are Ito processes. Made available in DSpace on 2011-05-07T11:59:20Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624313.pdf: 1508665 bytes, checksum: 319c96e239a6704c6759240300eeef2c (MD5) Previous issue date: 1995 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:35:11Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:13:46-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9624313 (UMI)AAI9624313 http://hdl.handle.net/2142/19178 eng Copyright 1995 Choi, Changsun Mathematics Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.