We study the $L^1$-smoothing properties for a broad class of semigroups arising from the ground state transformation of Schrödinger semigroups with confining potentials associated with non-local Lévy operators, for which (asymptotic) ultracontractivity and hypercontractivity fail. Our work is inspired by Talagrand's convolution conjecture in the discrete cube setting, as well as by subsequent developments on the classical Ornstein--Uhlenbeck semigroup. The estimates we provide exhibit a clear dependence on the potential and the Lévy measure defining the kinetic term operator, and they yield a description of the semigroups' action on $L^1$ in terms of Orlicz spaces. Our framework is quite general, encompassing fractional and relativistic Laplacians as kinetic operators. The results are illustrated by numerous examples demonstrating that the $L^1$-regularizing effects become stronger as $t \uparrow \infty$.
1.1. Motivation and introduction. Let µ be the standard Gaussian measure on R d , where d ∈ {1, 2, . . .}, and let {Q t : t ⩾ 0} be the classical Ornstein-Uhlenbeck semigroup, defined by
h(e -t x + 1 -e -2t y)µ(dy), for any admissible function h on R d . Nelson’s hypercontractivity theorem [39] states that for 1 < p < q < ∞, there exists t(p, q) > 0 such that for all t ⩾ t(p, q), the operator Q t : L p (µ) → L q (µ) is contractive. This implies that Q t has a regularizing effect on functions h ∈ L p (µ) whenever p > 1. Hypercontractivity is a powerful tool in the theory of operator semigroups, with numerous interesting and important connections and applications; see, e.g., [14,20,23] and additional references therein.
This raises a natural question: what happens when p = 1? Research in this direction was inspired by Talagrand’s convolution conjecture [50]. It was originally formulated in a discrete setting for the average of the heat semigroup on the hypercube but naturally extends to a much broader framework, including the Ornstein-Uhlenbeck semigroup and even the semigroups studied in the present paper. The conjecture states that, for every t > 0 there exists a function ψ t : [1, ∞) → [1, ∞) such that ψ t (u) → ∞ as u → ∞, and for every h ∈ L 1 (µ) with ∥h∥ L 1 (µ) = 1 and u ⩾ 1, the following holds:
.
We have Q t h(x)µ(dx) = h(x)µ(dx) and, by Markov’s inequality,
Therefore, the function ψ t (u) can be interpreted as the rate at which the action of the operator Q t smooths the functions h ∈ L 1 (µ). Talagrand observed that the best decay rate one can expect is ψ t (u) = c(t) √ log u, where the constant c(t) depends only on t.
The first positive answear to this problem for the Ornstein-Uhlenbeck semigroup was provided by Ball, Barthe, Bednorz, Oleszkiewicz, and Wolff [2], who proved that the estimate holds true with ψ t (u) = c(t, d) √ log u/ log log u, where the constant c(t, d) depends on the dimension d. Later, Eldan and Lee [20] improved the bound to ψ t (u) = c(t) log u/ log log u, where the constant c(t) depends only on t. Finally, Lehec [36] obtained the optimal estimate with ψ t (u) = c min{1, t} √ log u, where c is an absolute constant. We refer the reader to [20] for an excellent overview of this problem and methods, along with a discussion of various connections to related topics and works. See also the work of Gozlan, Madiman, Roberto, and Samson [22] for an interesting continuation of the research on deviation inequalities, and that of Gozlan, Li, Madiman, Roberto, and Samson [21] for extensions to more general diffusions, the M/M/∞ queue model, and the Laguerre semigroup in dimension one.
The aim of this paper is to explore the L 1 -smoothing effects for a broad class of semigroups {Q t : t ⩾ 0} arising from the ground state transformation of semigroups associated with Schrödinger operators H = -L + V . Here, L denotes a non-local Lévy operator, including, as particular cases, fractional and relativistic Laplacians, and V denotes a confining potential, i.e., V (x) → ∞ as |x| → ∞. These semigroups can be seen as natural non-local analogues, or even extensions, of classical diffusion semigroups such as the Ornstein-Uhlenbeck semigroup. We note that the stationary measures µ are no longer Gaussian, as they are now given by the squares of the ground state eigenfunctions φ 0 of the Schrödinger operators H. Hypercontractivity and ultracontractivity (corresponding to the strongest L 1 -to-L ∞ smoothing) of the semigroups {Q t : t ⩾ 0} have been extensively studied. Both properties are known to hold when the potential V grows sufficiently fast at infinity (see, for instance, Chen and Wang [11,12], Kulczycki and Siudeja [33], or Kaleta, Kwaśnicki, and Lőrinczi [27]). In contrast, the L 1 -smoothing properties in non-ultracontractive regime remain completely unexplored. To our knowledge, this work is the first to address this problem within the non-local Lévy framework. We present a unified approach that enables a systematic analysis and yields a solution for a broad class of L’s and V ’s.
Our main results are presented in Section 1.3, including the upper bound in Theorem 1.1 and the lower bounds in Theorems 1.2 and 1.3. While these estimates are very much in the spirit of Talagrand’s convolution conjecture, the structure of the rates we obtain differs significantly from those discussed above, reflecting the sharp dependence on the kinetic term Lévy operators and the confining potentials. Corollary 1.4 characterizes the range Q t L 1 (µ) in terms of Orlicz spaces.
To put our results into context and demonstrate some of their direct applications, we briefly discuss an example concerning the fractional Schrödinger operator with a power-logarithmic potential H = (-∆) a + log θ (1 + |x|), a ∈ (0, 1), θ > 0.
The long-time regularizing properties of the associated ground state-transformed semigroup {Q t : t ⩾ 0} exhibit a sharp dichotomy depending on the parameter θ. More precisely, it is known, se
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