Singular Integrals and Hardy Type Spaces for the Inverse Gauss Measure
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Singular Integrals and Hardy Type Spaces for the Inverse Gauss Measure Tommaso Bruno1 Received: 6 May 2020 / Accepted: 9 October 2020 © Mathematica Josephina, Inc. 2020
Abstract Let γ−1 be the absolutely continuous measure on Rn whose density is the reciprocal of a Gaussian and consider the weighted symmetric Laplacian A on L 2 (γ−1 ). We prove boundedness and unboundedness results for the purely imaginary powers and the first order Riesz transforms of A + λI , λ ≥ 0, from new Hardy spaces adapted to γ−1 to L 1 (γ−1 ). We also investigate their weak type (1, 1). Keywords Singular integrals · Endpoint results · Hardy spaces · Weak type · Gauss measure · Nondoubling measure Mathematics Subject Classification 42B20 · 42B30 · 42B35
1 Introduction Let n ∈ N and denote by γ−1 the absolutely continuous measure on Rn whose density is the reciprocal of a normalised Gaussian, i.e. dγ−1 (x) = π n/2 e|x| dx, 2
where dx is the Lebesgue measure on Rn . We call γ−1 the “inverse Gauss” measure. Consider the second-order differential operator A0 f (x) = − 21 f (x) − x · ∇ f (x),
f ∈ Cc∞ (Rn ),
The author acknowledges support by the Research Foundation – Flanders (FWO) through the postdoctoral grant 12ZW120N. He is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
B 1
Tommaso Bruno [email protected] Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium
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T. Bruno
which is nonnegative and symmetric on L 2 (γ−1 ). By a classical argument (see e.g. [25]) A0 is essentially self-adjoint; we denote by A its self-adjoint closure. In this paper, we prove endpoint results for the purely imaginary powers and the first order Riesz transforms associated with nonnegative translates of A. By these, we mean the operators (A + λI )iu , u ∈ R \ {0},
Rλ = ∇ (A + λI )−1/2 ,
(1.1)
for λ ≥ 0, respectively. As we shall see, the translates of A, rather than A itself, arise naturally. The operator A was first introduced in F. Salogni’s PhD thesis [23], where the semigroup e−t A is studied on L p (γ−1 ), p ∈ (1, ∞), and the weak type (1, 1) of its maximal operator is established. The interest in imaginary powers and Riesz transforms of A comes from different aspects. As pointed out in [23], the operator A can be seen as a restriction of the Laplace–Beltrami operator on a warped-product manifold whose Ricci tensor is unbounded from below. Since there is no theory available of singular integrals or Hardy spaces in such context, the study of imaginary powers and Riesz transforms of A may be a first step in its understanding. In addition to this, we may look at the operator A as the natural Laplacian of the weighted manifold (Rn , γ−1 ). On weighted Riemannian manifolds, there is a natural notion of curvature tensor known as Bakry–Emery curvature [1]. It was shown by Bakry [1] (see also [4]) that in the general setting of weighted Riemannian manifolds with Laplacia
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