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 Author Hou, Qi Title Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces URL http://hdl.handle.net/1813/67578 Publication Date 2019 Date Accessioned 2019-10-15 16:48:25 Degree PhD Discipline/Department Mathematics Degree Level doctoral University/Publisher Cornell University Abstract This thesis studies some qualitative properties of local weak solutions of the heat equation in Dirichlet spaces. Let $\left(X,\mathcal{E},\mathcal{F}\right)$\ be a Dirichlet space where $X$\ is a metric measure space, and $\left(\mathcal{E},\mathcal{F}\right)$\ is a symmetric, local, regular Dirichlet form on $L^2\left(X\right)$. Let $-P$\ and $\left(H_t\right)_{t>0}$\ denote the corresponding generator and semigroup. Consider the heat equation $\left(\partial_t+P\right)u=f$\ in $\mathbb{R}\times X$. 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 $\mathbb{R}^n$, 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 $\left(\partial_t+P\right)u=f$\ on some time-space cylinder $I\times \Omega$.\\ (i) If the time derivative of $f$\ is locally in $L^2\left(I\times \Omega\right)$, then the time derivative of $u$\ is a local weak solution to $\left(\partial_t+P\right)\partial_t u=\partial_t f$.\\ (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 $h\left(t,x,y\right)$, 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 $h\left(t,x,y\right)$. As a special case, on the infinite torus $\mathbb{T}^\infty$, local boundedness of $h\left(t,x,y\right)$\ implies automatically the continuity of $h\left(t,x,y\right)$, 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. Subjects/Keywords Dirichlet space; heat equation; heat kernel; heat semigroup; local weak solution; Mathematics Contributors Saloff-Coste, Laurent Pascal (chair); Healey, Timothy James (committee member); Cao, Xiaodong (committee member) Language en Country of Publication us Record ID handle:1813/67578 Repository cornell Date Retrieved 2020-09-07 Date Indexed 2020-09-09 Grantor Cornell University Issued Date 2019-08-30 00:00:00

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…example weak solutions satisfying the heat equations in the sense of forms) and the semigroup method (which can be used to treat for example solutions to the abstract heat equation (∂t − A) u = f where the solutions are viewed as…

…general approach we take is to utilize the heat semigroup to study the aforementioned properties of local weak solutions to heat equations from a hypoellipticity point of view. It differs from the classical hypoellipticity viewpoint in that it picks out…

…the heat semigroup as a special “fundamental solution” to the heat equation, and use it to study properties of general local weak solutions, while traditional studies of hypoellipticity in general treat all solutions equally. Some key ideas in our…

…product of functions still belongs to the domain of the “Laplacian”), and often lack notions like convolution. To overcome these complications we resort to the Dirichlet form and the heat semigroup. And the use of Dirichlet forms leads naturally to…

…weak solutions, as they no longer serve the role. “L∞ ” local boundedness, continuity, and time regularity - Chapter 4. The key result is the local boundedness of local weak solutions, given that the heat semigroup (Ht )t>0 associated with…

…local weak solutions, given the additional condition that the heat semigroup admits a density kernel continuous on some subset. We remark here that the local ultracontractivity condition implies the existence of a locally L2 density function, often…

…three types of examples to which our results apply or partially apply, and make more comparisons with existing studies. In short, the local point of view we take is essential in the possibility of imposing only local conditions on the heat semigroup to…

…choosing the most convenient semigroup and form to work with in showing properties of their corresponding local weak solutions, and the results then apply to all other heat equations with the same set of local weak solutions. In this spirit, in Chapters 5…