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^ϕƒ||L_(Ω) ≤ A||e^ϕ∆ƒ||Lp(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)
For the family of radial weights e^(|x|p), 1 <ρ <∞, we obtain an L^p(R^n) → L^q(R^n) inequality for 1/p-1/q = 2/n and 2n/(n +3)
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 one-dimensional weights e^(ϕ(xn)), for n ≥ 3, and when 1/p - 1/q = 2/n and 2n(n + 3)
