In this paper, we obtain certain sharp estimates for the maximal modulus of a rational function with prescribed poles. The proofs obtained results are based on new version Schwarz lemma regular functions which was suggested by Osserman. produce several inequalities polynomials as well.

Let $$G=(V,E)$$ be a finite graph (here V and E denote the set of vertices edges G respectively) $$M_G$$ centered Hardy–Littlewood maximal operator defined there. We find optimal value $$\mathbf{{C}}_{G,p}$$ such that inequality $$\begin{aligned} \mathrm{Var\,}_{p}M_{G}f\le \mathbf{C}_{G,p}\mathrm{Var\,}_{p}f \end{aligned}$$ holds for every $$f:V\rightarrow {\mathbb {R}},$$ where $$\mathrm{Var\...

X iv :m at h/ 02 10 08 4v 2 [ m at h. C A ] 1 3 D ec 2 00 2 Abstract. Recently Wolff [28] obtained a sharp L2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon ...

The purpose of this paper is to prove an essentially sharp L2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature. Fix d ≥ 3, denote variables in Rd by (x, xd) with x ∈ Rd−1, and let Γ = {x : xd = |x| and 1 ≤ xd ≤ 2}. Let Γ1 and Γ2 be disjoint conical subsets, i.e. Γi = {x ∈ Γ : x xd ∈ Ωi} where Ωi are disjoint closed subsets of the sphere...

where γ runs over the set of all geodesics with endpoints on ∂M . All potential fields dv given by (dv)ij = 1 2 (∇ivj +∇jvi) with v = 0 on ∂M belong to the kernel of I. The ray transform I is called s-injective if this is the only obstruction to injectivity, i.e., if If = 0 implies that f is potential. S-injectivity can only hold under certain assumptions on (M, g). A natural conjecture is that...