Citation
Evasius, Dean M. (1992) Carleman inequalities with convex weights. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4v8qvv71. https://resolver.caltech.edu/CaltechTHESIS:09092011152345667
Abstract
In this thesis we show that if n ≥ 2, and ϕ is a convex function on the bounded convex domain Ω, then there is a constant A = A(n,p,q,Ω) such that e^ϕƒL_(Ω) ≤ Ae^ϕ∆ƒLp(Ω) holds for all ƒ Є C(^∞_0)(Ω), and for the following values of p and q: p = n/2 and q < 2n/(n  3) when n ≥ 3, and p > 1 and q < ∞ when n = 2.
For the one parameter family of weights {e^(tϕ)}_(t ≥ 1 ) where ϕ is essentially uniformly convex on a bounded domain Ω, we prove an L^p(Ω) → L^q(Ω) inequality for 1/p 1/q ≤ 2/n and 2n/(n + 3) < p ≤ q < 2n/(n  3), n ≥ 3, (1 < p ≤ q < ∞ for n = 2).
For the family of radial weights e^(xp), 1 < ρ < ∞, we obtain an L^p(R^n) → L^q(R^n) inequality for 1/p1/q = 2/n and 2n/(n +3) < p ≤ q < 2n/(n  3), n ≥ 3. For 2 ≤ ρ < ∞, this can be improved to 1/p  1/q ≤ 2/n and 2n/(n  3) < p ≤ q < 2n/(n  3) when n ≥ 3. If n = 2, the valid range is 1 < p ≤ q < ∞.
Finally, if ϕ is any convex function on R, we obtain an L^P(R^n) → L^q(R^n) Carleman inequality for the family of onedimensional weights e^(ϕ(xn)), for n ≥ 3, and when 1/p  1/q = 2/n and 2n(n + 3) < p < q < 2n/(n  3).
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  4 May 1992 
Record Number:  CaltechTHESIS:09092011152345667 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:09092011152345667 
DOI:  10.7907/4v8qvv71 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6661 
Collection:  CaltechTHESIS 
Deposited By:  John Wade 
Deposited On:  12 Sep 2011 21:26 
Last Modified:  16 Apr 2021 22:29 
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