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You searched for subject:( heat semigroup). Showing records 1 – 3 of 3 total matches.

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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

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 L2 ≤ ft(X)). Let -P and  ≤ ft(Ht))t>0 denote the corresponding generator and semigroup. Consider the heat equation  ≤ ft(\partialt+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 ℝn, 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(\partialt+P))u=f on some time-space cylinder I ×  Ω.\ (i) If the time derivative of f is locally in L2 ≤ ft(I ×  Ω)), then the time derivative of u is a local weak solution to  ≤ ft(\partialt+P))\partialt u=\partialt f.\ (ii) If the semigroup Ht 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 Ht 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 (6th 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 (16th 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 (7th 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


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

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 Cd and Laplace-Beltrami operator on the group SU(2) are obtained using their Lpestimates 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 LP-spaces. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted Lp-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 Lp ( Rd,〖 (φiρ (x ))〗2 dx) where φiρ 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

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

APA (6th 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 (16th Edition):

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Doctoral Dissertation, Indian Institute of Science. Accessed September 26, 2020. http://etd.iisc.ac.in/handle/2005/2958.

MLA Handbook (7th Edition):

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Web. 26 Sep 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 Sep 26]. 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

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

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

APA (6th 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 (16th Edition):

Rezvani, Sepideh. “Approximating rotation algebras and inclusions of C*-algebras.” 2017. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 26, 2020. http://hdl.handle.net/2142/97307.

MLA Handbook (7th Edition):

Rezvani, Sepideh. “Approximating rotation algebras and inclusions of C*-algebras.” 2017. Web. 26 Sep 2020.

Vancouver:

Rezvani S. Approximating rotation algebras and inclusions of C*-algebras. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2017. [cited 2020 Sep 26]. 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

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