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Hou, Qi.
Rough Hypoellipticity for Local Weak Solutions to the *Heat* Equation in Dirichlet Spaces.

Degree: PhD, Mathematics, 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: Saloff-Coste, Laurent Pascal (chair), Healey, Timothy James (committee member), Cao, Xiaodong (committee member).*

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 )…

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APA (6^{th} Edition):

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

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

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces.” 2019. Doctoral Dissertation, Cornell University. Accessed September 26, 2020. http://hdl.handle.net/1813/67578.

MLA Handbook (7^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces.” 2019. Web. 26 Sep 2020.

Vancouver:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces. [Internet] [Doctoral dissertation]. Cornell University; 2019. [cited 2020 Sep 26]. Available from: http://hdl.handle.net/1813/67578.

Council of Science Editors:

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