Fractional Laplacian, homogeneous Sobolev spaces and their realizations
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Fractional Laplacian, homogeneous Sobolev spaces and their realizations Alessandro Monguzzi1 · Marco M. Peloso2 · Maura Salvatori2 Received: 14 October 2019 / Accepted: 15 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the fractional Laplacian and the homogeneous Sobolev spaces on ℝd , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way, we also prove some properties of the fractional Laplacian. Keywords Homogeneous Sobolev spaces · Fractional Laplacian Mathematics Subject Classification 46E35 · 42B35
1 Introduction and statement of the main results The goal of this paper is to clarify a point that in our opinion has been overlooked in the literature. Classically, the homogeneous Sobolev spaces and the fractional Laplacian are defined in two different ways. In one case, we consider the Laplacian Δ as densely defined, self-adjoint and positive on L2 (ℝd ) , and following Komatsu [15], it is possible to define the fractional powers Δs∕2 by means of the spectral theorem, s > 0 . Denoting All authors are partially supported by the grant PRIN 2015 Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis and are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). * Marco M. Peloso [email protected] Alessandro Monguzzi [email protected] Maura Salvatori [email protected] 1
Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano–Bicocca, Via R. Cozzi 55, 20126 Milan, Italy
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Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milan, Italy
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by S the space of Schwartz functions, one observes that for p ∈ (1, ∞) , ‖Δs∕2 𝜑‖Lp (ℝn ) is a norm on S . The homogeneous Sobolev space is the space Ẇ s,p defined as the closure of S in such a norm. The second definition is modeled on the classical Littlewood–Paley decomposition of function spaces (see details below) and gives rise to an operator that we will denote by Δ̇ s∕2 , acting on spaces of tempered distributions modulo polynomials. Below, we describe the latter approach and show how these two definitions are related to each other, by showing that they admit the same, explicit realization. We mention that the analysis involved by the fractional Laplacian Δs has drawn great interest in the latest years, beginning with the groundbreaking papers [9, 10], see also the recent papers [3, 4, 8, 14, 18, 20, 24, 25]. We also mention that we were led to consider this problem while working on spaces of entire functions of exponential type whose fractional Laplacian is Lp on the real line, [19]. Let S denote the space of Schwar
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