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1.
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 )…

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. (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 October 20, 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. 20 Oct 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 Oct 20]. 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

Indian Institute of Science

2.
Boggarapu, Pradeep.
Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of *Heat* Semigroups.

Degree: PhD, Faculty of Science, 2018, Indian Institute of Science

URL: http://etd.iisc.ac.in/handle/2005/2958

This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the sub-Laplacian on Heisenberg group, special Hermite operator on C^{d} and Laplace-Beltrami operator on the group SU(2) are obtained using their L^{pestimates} and by making use of a lemma of Herz and Riviere along with an idea of Rubio de Francia. Applying these results to functions expanded in terms of spherical harmonics, we deduce certain vector valued inequalities for sequences of operators defined in terms of radial parts of the corresponding operators.
In the second part, we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian ∆_κ on weighted L^{P-spaces}. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted L^{p-spaces}, we only have partial results, but in the case of the heat semigroup generated by the standard Laplacian, a complete picture of the chaotic behavior is obtained on the spaces L^{p} ( R^{d},〖 (φ_{i}ρ (x ))〗^{2} dx) where φ_{i}ρ the Euclidean spherical function is. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.
In the last part, we study mixed norm estimates for the Cesáro means associated with Dunkl-Hermite expansions on〖 R〗^{d}. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator)〖 H〗_κ:=-Δ_κ+|x|^{2}. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesáro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of Three Lines Lemma.
*Advisors/Committee Members: Thangavelu, Sundaram (advisor).*

Subjects/Keywords: Dunkl Transforms; Dunkl Heat Semigroups; Differential Operators; Riesz Transforms; Dunkl-Laplacian; Dunkl Harmonic Oscillator; Heisenberg Group; Heat Semigroups Chaotic Behavior; Dunkl-Hermite Expansions; Dunkl-Hermite Operators; Dunkl Operators; Cesaro Means; Laguerre Function Expansions; Dunkl Heat Semigroup; Mathematics

Record Details Similar Records

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

APA (6^{th} Edition):

Boggarapu, P. (2018). Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups. (Doctoral Dissertation). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/2958

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

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Doctoral Dissertation, Indian Institute of Science. Accessed October 20, 2020. http://etd.iisc.ac.in/handle/2005/2958.

MLA Handbook (7^{th} Edition):

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Web. 20 Oct 2020.

Vancouver:

Boggarapu P. Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups. [Internet] [Doctoral dissertation]. Indian Institute of Science; 2018. [cited 2020 Oct 20]. Available from: http://etd.iisc.ac.in/handle/2005/2958.

Council of Science Editors:

Boggarapu P. Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups. [Doctoral Dissertation]. Indian Institute of Science; 2018. Available from: http://etd.iisc.ac.in/handle/2005/2958

University of Illinois – Urbana-Champaign

3. 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

❌

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 October 20, 2020. http://hdl.handle.net/2142/97307.

MLA Handbook (7^{th} Edition):

Rezvani, Sepideh. “Approximating rotation algebras and inclusions of C*-algebras.” 2017. Web. 20 Oct 2020.

Vancouver:

Rezvani S. Approximating rotation algebras and inclusions of C*-algebras. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2017. [cited 2020 Oct 20]. 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