Riesz transforms on connected sums
نویسنده
چکیده
On the Euclidean space, it is well known that the Riesz transform has also a bounded extension L(M) → L(M ;TM) for any p ∈]1,∞[. However, this is not a general feature of the Riesz transform on complete Riemannian manifolds, as the matter of fact, on the connected sum of two copies of the Euclidean space R , the Riesz transform is not bounded on L for any p ∈ [n,∞[∩]2,∞[ ([9, 7]). It is of interest to figure out the range of p for which T extends to a bounded map L(M) → L(M ;T ∗M). The main result of [7] answered to this question for manifolds with Euclidean ends :
منابع مشابه
Weighted Divisor Sums and Bessel Function Series, Iv
Abstract. One fragment (page 335) published with Ramanujan’s lost notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second,...
متن کاملRiesz transforms through reverse Hölder and Poincaré inequalities
We study the boundedness of Riesz transforms in L for p > 2 on a doubling metric measure space endowed with a gradient operator and an injective, ω-accretive operator L satisfying Davies-Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on L for p in some interval [2, 2 + ε), and that L gradient estimates ...
متن کاملHigher order Riesz transforms for Hermite expansions
In this paper, we consider the Riesz transform of higher order associated with the harmonic oscillator [Formula: see text], where Δ is the Laplacian on [Formula: see text]. Moreover, the boundedness of Riesz transforms of higher order associated with Hermite functions on the Hardy space is proved.
متن کاملRiesz transforms and Lie groupsof polynomial
Let G be a Lie group of polynomial growth. We prove that the second-order Riesz transforms on L 2 (G ; dg) are bounded if, and only if, the group is a local direct product of a compact group and a nilpotent group, in which case the transforms of all orders are bounded.
متن کاملBellman function, Littlewood-Paley estimates and asymptotics for the Ahlfors-Beurling operator in L(C)
Estimation of L norms of Fourier multipliers is known to be hard. It is usually connected to some interesting types of PDE, see several such PDE for several Fourier multipliers on the line in a recent paper of Kalton and Verbitsky [13]. Sometimes, but much more rarely, one can establish sharp L estimates for Fourier multipliers in several variables. Riesz transforms are examples of success. The...
متن کامل