New Riemannian manifolds with L p -unbounded Riesz transform for p > 2
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Mathematische Zeitschrift
New Riemannian manifolds with Lp -unbounded Riesz transform for p > 2 Alex Amenta1 Received: 31 July 2017 / Accepted: 28 February 2020 © The Author(s) 2020
Abstract We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on L p (M) for all p > 2. This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.
1 Introduction Consider a Riemannian manifold M with gradient ∇ and Laplace–Beltrami operator Δ. The Riesz transform ∇Δ−1/2 , with Δ−1/2 defined via the spectral theorem, maps L 2 (M) boundedly to the space of square integrable vector fields L 2 (M; T M). Much attention has been given to the question of whether this operator extends to a bounded map from L p (M) to L p (M; T M) for p = 2, or equivalently, whether the estimate (R p ) :
|∇ f | p Δ1/2 f p
for all f ∈ Cc∞ (M)
holds. It is conjectured that for p ∈ (1, 2) the estimate (R p ) holds whenever M is complete, with implicit constant depending only on p; in [1] it is shown that the failure of this uniformity among manifolds of a fixed dimension would imply the existence of a manifold for which (R p ) fails. One is naturally led to consider also the ‘reverse’ estimate (R R p ) :
Δ1/2 f p |∇ f | p
for all f ∈ Cc∞ (M).
A duality argument shows that for p ∈ (1, ∞), (R p ) implies (R R p ), where p = p/( p − 1) is the Hölder conjugate of p. If (R p ) and (R R p ) both hold, then we have a norm equivalence |∇ f | p Δ1/2 f p , p which says that the homogeneous Sobolev space W˙ 1 (M) may be defined either via the gradient or via the square root of the Laplace–Beltrami operator. Generally (R p ) holds only for some interval of p ∈ (1, ∞) including 2, and proving (R p ) presents different difficulties depending on whether p < 2 or p > 2. When 1 < p < 2,
B 1
Alex Amenta [email protected] Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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A. Amenta
(R p ) is known to follow from the volume doubling property and Gaussian or sub-Gaussian heat kernel upper estimates [9,10] (see also [15] for examples which do not satisfy such kernel estimates). The volume doubling property and an appropriately scaled L 2 -Poincaré inequality imply (R p ) for some p > 2 [2]. In [3] this is linked with gradient estimates for the heat kernel, and in [5] the L 2 -Poincaré inequality is replaced by a relative Faber–Krahn inequality and a reverse Hölder inequality. Some manifolds for which (R p ) fails for some p > 2 are known. If M is the connected sum of two copies of Rn \ B(0, 1) with n ≥ 3 —or more generally, an n-dimensional manifold with at least two (and finitely many) Euclidean ends—(R p ) holds if and only if p ∈ (1, n) [7,10]. Similar results are known for conical manifolds [14] and for 2-hyperbolic, p-parabolic manifolds with at least two ends [6]. The most relevant examples to this article are Vicsek manifolds, which are ‘thickenings’ of Vicsek graphs (pictu
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