Ornstein-Uhlenbeck semigroup maximal operator on weighted Lp space
Abstract
Let (ℋt)t ≥ 0 denote the standard Ornstein-Uhlenbeck semigroup over ambient space ℝd equipped with Gaussian measure dγ(x) = e−|x|2 dx. Further let ℋ*(f)(x) = supt > 0 ℋt(|f|)(x) denote the associated maximal operator. Since (ℋt)t ≥ 0 is a Markovian semigroup over (ℝd,dγ(x)), it is well known that ℋ* is bounded on Lp(ℝd,dγ(x)) for all 1 < p ≤ ∞. We show that ℋ* is also bounded on the weighted space Lp(ℝd,ω(x)e−p/2|x|2dx) for 1 < p <∞, for weights ω : ℝd → (0,∞) belonging to a certain Muckenhoupt type class Aαp, where α ∈ [0,1). Essentially, a weight ω belongs to Aαp if and only if ω(Q) ω−1/(p−1)(Q)p−1 ≤ C |Q|p for all cubes Q contained in any reference cube Nα(Rx) ⊆ ℝd, where the well chosen family (Nα(Rx))x ∈ ℝd covers ℝd, Nα(Rx) contains x and the side length of Nα(Rx) is comparable to 1/max(1,|x|)α, with constants depending only on d. The class Aαp contains the usual Muckenhoupt class. We also use the weight class Aploc from \cite[Definition 2.2]{B} defined in a similar way as Aαp but with dyadic subcubes contained in some N(R) built over the Gaussian dyadic grid (see below), and we show that the local part supt >0ℋt(|f|χN(Rx))(x) is bounded on Lp(ℝd,ω(x)e−p/2|x|2dx) if and only if ω ∈ Alocp.
Related
- PSEUDOSPECTRAL PHENOMENA PREVENTING THE ANALYTICITY SECTOR OF A ORNSTEIN-UHLENBECK SEMIGROUP TO BE RULED BY THE SPECTRAL ANGLE OF ITS GENERATOR
- Spectral and mapping properties of the inverse Ornstein--Uhlenbeck operator
- The Ornstein−Uhlenbeck process on 𝒫2 with a volatility operator
- Sharp weighted estimates for the multilinear maximal operator
- Weighted boundedness of maximal operator associated with invertible linear transformations on variable Lebesgue spaces
BibTeX
@misc{kriegler2026ornsteinuhlenbeck,
title = {{Ornstein-Uhlenbeck semigroup maximal operator on weighted $L^p$ space}},
author = {Christoph Kriegler and Jérémie Moukambi},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28571},
eprint = {2609.28571},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05757041v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ornstein-Uhlenbeck semigroup maximal operator on weighted L^p space
AU - Christoph Kriegler
AU - Jérémie Moukambi
PY - 2026
DA - 2026/09/23
PB - HAL
DO - 10.48550/arxiv.2609.28571
UR - https://hal.science/hal-05757041v1
AB - Let (ℋ_t)_t ≥ 0 denote the standard Ornstein-Uhlenbeck semigroup over ambient space ℝ^d equipped with Gaussian measure dγ(x) = e^−|x|^2 dx. Further let ℋ^*(f)(x) = sup_t > 0 ℋ_t(|f|)(x) denote the associated maximal operator. Since (ℋ_t)_t ≥ 0 is a Markovian semigroup over (ℝ^d,dγ(x)), it is well known that ℋ^* is bounded on L^p(ℝ^d,dγ(x)) for all 1 < p ≤ ∞. We show that ℋ^* is also bounded on the weighted space L^p(ℝ^d,ω(x)e^−p/2|x|^2dx) for 1 < p <∞, for weights ω : ℝ^d → (0,∞) belonging to a certain Muckenhoupt type class A^α_p, where α ∈ [0,1). Essentially, a weight ω belongs to A^α_p if and only if ω(Q) ω^−1/(p−1)(Q)^p−1 ≤ C |Q|^p for all cubes Q contained in any reference cube N_α(R_x) ⊆ ℝ^d, where the well chosen family (N_α(R_x))_x ∈ ℝ^d covers ℝ^d, N_α(R_x) contains x and the side length of N_α(R_x) is comparable to 1/max(1,|x|)^α, with constants depending only on d. The class A^α_p contains the usual Muckenhoupt class. We also use the weight class A_p^loc from \cite[Definition 2.2]{B} defined in a similar way as A^α_p but with dyadic subcubes contained in some N(R) built over the Gaussian dyadic grid (see below), and we show that the local part sup_t >0ℋ_t(|f|χ_N(R_x))(x) is bounded on L^p(ℝ^d,ω(x)e^−p/2|x|^2dx) if and only if ω ∈ A^loc_p.
ER -