Boundedness of multi-parameter pseudo-differential operators on multi-parameter Lipschitz spaces
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Boundedness of multi-parameter pseudo-differential operators on multi-parameter Lipschitz spaces Shaoyong He1 · Jiecheng Chen1 Received: 1 July 2020 / Revised: 28 July 2020 / Accepted: 3 August 2020 © Springer Nature Switzerland AG 2020
Abstract The main purpose of this paper is to introduce the multi-parameter Lipschitz spaces and characterize it via the Littlewood–Paley theory. As an application to the multiparameter Lipschitz spaces, we derive the boundedness of multi-parameter pseudodifferential operators on multi-parameter Lipschitz spaces. Keywords Multi-parameter Lipschitz spaces · Littlewood–Paley theory · Multi-parameter pseudo-differential operators Mathematics Subject Classification 42B35 · 42B25 · 42B20
1 Introduction Since the celebrated work of L. Hörmander on linear pseudo-differential operators, extensive works have been done in the past decades. As is well-known, pseudodifferential operators play an important role in many branches of mathematics, e.g., partial differential equations, geometric analysis, harmonic analysis, several complex variables, etc. More precisely, pseudo-differential operators are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂¯ problem, see, e.g., [27,28,31,32,42,44], etc. The extensive applications of it further promote the position of Fourier integral operators in harmonic analysis(see [10,29,43]).
This research was funded by National Natural Science Foundation of China (Grant No. 11671363).
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Shaoyong He [email protected] Jiecheng Chen [email protected]
1
Department of Mathematics, Zhejiang Normal University, Jinhua, China
S. He, J. Chen
Let m, ρ and δ be real numbers. Following [28], a symbol σ in Hörmander class m is a smooth function defined on Rn × Rn , satisfying Sρ,δ β
|∂xα ∂ξ σ (x, ξ )| ≤ Cα,β (1 + |ξ |)m+δ|α|−ρ|β| for all multi-indices α, β and some positive constants Cα,β depending only on α, β. The corresponding pseudo-differential operator Tσ is given by Tσ f (x) =
Rn
σ (x, ξ ) fˆ(ξ )ei x·ξ dξ
for f ∈ S(Rn ), where fˆ is the Fourier transform of f , x ∈ Rn . The symbol S(Rn ) has its usual meaning, a Schwartz class of rapidly decreasing smooth functions in Rn . We denote its dual space by S (Rn ). Given a Schwartz function f on Rn , f S (Rn ) denotes its seminorm. As far as the L p boundedness is concerned, Hörmander [27] proved that the oper0 are L 2 bounded when 0 ≤ δ < ρ ≤ 1. Subsequently, ators with symbols in Sρ,δ Calde´ron and Vaillancourt [1] established the L 2 boundedness when 0 ≤ δ = ρ < 1, m = 0 by using an almost orthogonality technique in a Hilbert space. C. Fefferman [11] further generalized this result to L p boundedness when 1 < p < ∞ for m with 0 ≤ δ < ρ ≤ 1 and operators with symbols in Sρ,δ 1 1 m ≤ n − (ρ − 1). p 2
(1)
In addition, the result of C. Fefferman is sharp in the sense that if m > n| 1p − 21 |(ρ −1), then the L p boundedness fails. At the extreme values of p, p = 1, ∞, it is natural to hope that the Hardy spaces and BMO or Lipchitz spaces boundedn
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