Abstract
AbstractLet $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.
Highlights
Consider the weighted manifold R(n,v), defined as Rn with the Euclidean distance and the measure d μ(x) = e2⟨v,x⟩dx
In [3], Anker proved the weak type (1, 1) of first- and second-order Riesz transforms in a symmetric space of the non-compact type
The exponents cannot be replaced by any smaller numbers
Summary
We will consider Riesz transforms and Littlewood–Paley–Stein functions of any order in R(n,v) These operators are defined and studied in many general settings, such as Lie groups, symmetric spaces, and other Riemannian manifolds. In [3], Anker proved the weak type (1, 1) of first- and second-order Riesz transforms in a symmetric space of the non-compact type. This case is subtler because of the spectral gap. For k ≥ 2, hk and Hk are not of weak type (1, 1); there exists a constant C = C(v, k) such that for all μ In these two estimates, the exponents cannot be replaced by any smaller numbers. By A ≲ B, we mean A ≤ CB with such a C (we say that A is controlled by B), and A ∼ B stands for A ≤ CB and B ≤ CA
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