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1.
Hou, Qi.
Rough Hypoellipticity for Local Weak Solutions to the *Heat* Equation in Dirichlet Spaces
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Degree: 2019, Cornell University

URL: http://hdl.handle.net/1813/67578

This thesis studies some qualitative properties of local weak solutions of the heat equation in Dirichlet spaces. Let ≤ ft(X,𝓔,𝓕)) be a Dirichlet space where X is a metric measure space, and ≤ ft(𝓔,𝓕)) is a symmetric, local, regular Dirichlet form on L^{2} ≤ ft(X)). Let -P and ≤ ft(H_{t}))_{t>0} denote the corresponding generator and semigroup. Consider the heat equation ≤ ft(\partial_{t+P}))u=f in ℝ × X. Examples of such heat equations include the ones associated with (i) Dirichlet forms associated with uniformly elliptic, second order differential operators with measurable coefficients on ℝ^{n}, and Dirichlet forms on fractal spaces;\ (ii) Dirichlet forms associated with product diffusions and product anomalous diffusions on infinite products of compact metric measure spaces, including the infinite dimensional torus, and the infinite product of fractal spaces like the Sierpinski gaskets.\ We ask the following qualitative questions about local weak solutions to the above heat equations, which in spirit are generalizations of the notion of hypoellipticity: Are they locally bounded? Are they continuous? Is the time derivative of a local weak solution still a local weak solution? Under some hypotheses on existence of cutoff functions with either bounded gradient or bounded energy, and sometimes additional hypotheses on the semigroup, we give (partially) affirmative answers to the above questions. Some of our key results are as follows. Let u be a local weak solution to ≤ ft(\partial_{t+P}))u=f on some time-space cylinder I × Ω.\ (i) If the time derivative of f is locally in L^{2} ≤ ft(I × Ω)), then the time derivative of u is a local weak solution to ≤ ft(\partial_{t+P}))\partial_{t} u=\partial_{t} f.\ (ii) If the semigroup H_{t} is locally ultracontractive, and satisfies some Gaussian type upper bound, and if f is locally bounded, then u is locally bounded.\ (iii) Besides satisfying local contractivity and some Gaussian type upper bound, if the semigroup H_{t} further admits a locally continuous kernel h ≤ ft(t,x,y)), then u is locally continuous.\ (iv) If the semigroup is locally ultracontractive and satisfies some Gaussian type upper bound, then it admits a locally bounded function kernel h ≤ ft(t,x,y)). As a special case, on the infinite torus \mathbb{T}^∞, local boundedness of h ≤ ft(t,x,y)) implies automatically the continuity of h ≤ ft(t,x,y)), and hence of all local weak solutions.\ (v) The needed Gaussian type upper bounds can often be derived from the ultracontractivity conditions. We also discuss such implications under existence of cutoff functions with bounded gradient or bounded energy.\ The results presented in this thesis are joint work with Laurent Saloff-Coste.
*Advisors/Committee Members: Healey, Timothy James (committeeMember), Cao, Xiaodong (committeeMember).*

Subjects/Keywords: Dirichlet space; heat equation; heat kernel; heat semigroup; local weak solution; Mathematics

…the *heat* equations in the sense of forms) and the *semigroup* method (which can be… …to utilize the
*heat* *semigroup* to study the aforementioned properties of local weak… …hypoellipticity viewpoint in that it picks out the *heat* *semigroup* as a special
“fundamental solution” to… …Dirichlet form and the *heat* *semigroup*. And the use of
Dirichlet forms leads naturally to the… …solutions, given that the *heat* *semigroup*
(Ht )t>0 associated with (E, F )…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Hou, Q. (2019). Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces . (Thesis). Cornell University. Retrieved from http://hdl.handle.net/1813/67578

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces .” 2019. Thesis, Cornell University. Accessed August 10, 2020. http://hdl.handle.net/1813/67578.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces .” 2019. Web. 10 Aug 2020.

Vancouver:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces . [Internet] [Thesis]. Cornell University; 2019. [cited 2020 Aug 10]. Available from: http://hdl.handle.net/1813/67578.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces . [Thesis]. Cornell University; 2019. Available from: http://hdl.handle.net/1813/67578

Not specified: Masters Thesis or Doctoral Dissertation

University of Illinois – Urbana-Champaign

2. Rezvani, Sepideh. Approximating rotation algebras and inclusions of C*-algebras.

Degree: PhD, Mathematics, 2017, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/97307

In the first part of this thesis, we will follow Kirchberg’s categorical perspective to establish new notions of WEP and QWEP relative to a C∗-algebra, and develop similar properties as in the classical WEP and QWEP. Also we will show some examples of relative WEP and QWEP to illustrate the relations with the classical cases.
The focus of the second part of this thesis is the approximation of rotation algebras in the quantum Gromov–Hausdorff distance. We introduce the completely bounded quantum Gromov–Hausdorff distance and show that for even dimensions, the higher dimensional rotation algebras can be approximated by matrix algebras in this sense. Finally, we show that for even dimensions, matrix algebras converge to the rotation algebras in the strongest form of Gromov–Hausdorff distance, namely in the sense of Latrémolière’s Gromov–Hausdorff propinquity.
*Advisors/Committee Members: Junge, Marius (advisor), Boca, Florin (Committee Chair), Ruan, Zhong-Jin (committee member), Oikhberg, Timur (committee member).*

Subjects/Keywords: C*-algebras; Weak expectation property (WEP); Quotient weak expectation property (QWEP); A-WEP; A-QWEP; Relatively weak injectivity; Order-unit space; Noncommutative tori; Compact quantum metric space; Conditionally negative length function; Heat semigroup; Poisson semigroup; Rotation algebra; Continuous field of compact quantum metric spaces; Gromov–Hausdorff distance; Completely bounded quantum Gromov–Hausdorff distance; Gromov–Hausdorff propinquity

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Rezvani, S. (2017). Approximating rotation algebras and inclusions of C*-algebras. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/97307

Chicago Manual of Style (16^{th} Edition):

Rezvani, Sepideh. “Approximating rotation algebras and inclusions of C*-algebras.” 2017. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed August 10, 2020. http://hdl.handle.net/2142/97307.

MLA Handbook (7^{th} Edition):

Rezvani, Sepideh. “Approximating rotation algebras and inclusions of C*-algebras.” 2017. Web. 10 Aug 2020.

Vancouver:

Rezvani S. Approximating rotation algebras and inclusions of C*-algebras. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2017. [cited 2020 Aug 10]. Available from: http://hdl.handle.net/2142/97307.

Council of Science Editors:

Rezvani S. Approximating rotation algebras and inclusions of C*-algebras. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2017. Available from: http://hdl.handle.net/2142/97307