On necessary and sufficient conditions for 𝐿^{𝑝}-estimates of Riesz transforms associated to elliptic operators on ℝⁿ and related estimates

Abstract

This article focuses on L p estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the L p estimates. It appears that the case p < 2 already treated earlier is radically different from the case p > 2 which is new. We thus recover in a unified and coherent way many L p estimates and give further applications. The key tools from harmonic analysis are two criteria for L p boundedness, one for p < 2 and the other for p > 2 but in ranges different from the usual intervals (1, 2) and (2, ∞).

Hardy and BMO spaces associated to divergence form elliptic operators

“…The proof closely follows an analogous argument for vertical square function (see [4]). We omit the details.…”

“…We note that it has been shown in [4] that the intervals of p ≤ 2 such that the heat semigroup and Riesz transform are L p -bounded have the same interior. In the sequel, we shall denote by (p L , p L ) the interior of the interval of L p boundedness of the semigroup, and we recall [4] that p L > 2n/(n − 2).…”

“…Similar statements apply to the heat semigroup e −tL (for which L p boundedness may fail also for p finite, but sufficiently large), as well as to the square function (see (1.14) below). This failure of L p bounds for some p ∈ (1, 2) (and in (2, ∞)), which relies on counterexamples built in [25], [5], [16], along with some observations in [4], illustrates the significant difference between the operators associated to L and those arising in the classical Calderón-Zygmund theory. We note that it has been shown in [4] that the intervals of p ≤ 2 such that the heat semigroup and Riesz transform are L p -bounded have the same interior.…”

“…This failure of L p bounds for some p ∈ (1, 2) (and in (2, ∞)), which relies on counterexamples built in [25], [5], [16], along with some observations in [4], illustrates the significant difference between the operators associated to L and those arising in the classical Calderón-Zygmund theory. We note that it has been shown in [4] that the intervals of p ≤ 2 such that the heat semigroup and Riesz transform are L p -bounded have the same interior. In the sequel, we shall denote by (p L , p L ) the interior of the interval of L p boundedness of the semigroup, and we recall [4] that p L > 2n/(n − 2).…”

Hardy Spaces of Differential Forms on Riemannian Manifolds

“…Fortunately, there is a weaker notion of Gaussian decay, which holds on any complete Riemannian manifold, namely the notion of L 2 off-diagonal estimates, as introduced by Gaffney [27]. This notion has already proved to be a good substitute of Gaussian estimates for such questions as the Kato square root problem or L p -bounds for Riesz transforms when dealing with elliptic operators (even in the Euclidean setting) for which Gaussian estimates do not hold (see [1,4,9] in the Euclidean setting, and [2] in a complete Riemannian manifold). We show in the present work that a theory of Hardy spaces of differential forms can be developed under such a notion.…”