Weak type (p,p) bounds for Schrödinger groups via generalized Gaussian estimates

2020/07/03 by Zhijie Fan, Fan, Zhijie
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2007.01468

openalex publication_date 2020/07/03 · openalex created_date 2020/07/10 · openalex updated_date 2026/07/28

Abstract

Let L be a non-negative self-adjoint operator acting on L²(X), where X is a space of homogeneous type with a dimension n. Suppose that the heat operator e^-tL satisfies the generalized Gaussian (p₀, p'₀)-estimates of order m for some 1≤ p₀ < 2. It is known that the operator (I+L)^-s e^itL is bounded on L^p(X) for s≥ n|1/ 2-1/p| and p∈ (p₀, p₀') (see for example, \citeBlunck2, BDN, CCO, CDLY, DN, Mi1). In this paper we study the endpoint case p=p₀ and show that for s₀= n|1\over 2-1\over p₀|, the operator (I+L)^-s₀e^itL is of weak type (p₀,p₀), that is, there is a constant C>0, independent of t and f so that μ(\x: |(I+L)^-s₀e^itL f(x)|gt;α\ )≤ C (1+|t|)^n(1 - p₀\over 2) ( ‖f‖_p0 \over α )^p₀ , t∈\mathbb R for α>0 when μ(X)=∞, and α>(‖f‖_p₀/μ(X) )^p₀ when μ(X)

Citations

• Estimates for translation invariant operators in Lp spaces
• Hp spaces of several variables

Related

• Sharp endpoint L^p estimates for Schrödinger groups
• Weak type (1,1) bounds for Schrödinger groups
• Sharp L^p estimates for Schrödinger groups on spaces of homogeneous type
• Sharp endpoint estimates for Schrödinger groups on Hardy spaces
• Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers