An Lq(Lp)-theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps

• Published in: Stochastic partial differential equations : analysis and computations 2024-09, Vol.12 (3), p.1439-1491
• Main Authors: Kang, Jaehoon, Park, Daehan
• Format: Article
• Language: English
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Differential equations; Function space; Mathematics; Mathematics and Statistics; Numerical Analysis; Operators (mathematics); Partial Differential Equations; Probability Theory and Stochastic Processes; Relativity; Statistical Theory and Methods; Stochastic processes
• ISSN: 2194-0401; 2194-041X
• DOI: 10.1007/s40072-023-00309-6

We investigate an L q ( L p ) -regularity ( 1 < p , q < ∞ ) theory for space-time nonlocal equations of the type ∂ t α u = L u + f . Here, ∂ t α is the Caputo fractional derivative of order α ∈ ( 0 , 1 ) and L is an integro-differential operator L u ( x ) = ∫ R d u ( x ) - u ( x + y ) - ∇ u ( x ) · y 1 | y | ≤ 1 j d ( | y | ) d y which is the infinitesimal generator of an isotropic unimodal Lévy process. We assume that the jumping kernel j d ( r ) is comparable to r - d ℓ ( r - 1 ) , where ℓ is a continuous function satisfying C 1 R r δ 1 ≤ ℓ ( R ) ℓ ( r ) ≤ C 2 R r δ 2 for 1 ≤ r ≤ R < ∞ , where 0 ≤ δ 1 ≤ δ 2 < 2 . Hence, ℓ can be slowly varying at infinity. Our result covers L whose Fourier multiplier Ψ ( ξ ) satisfies Ψ ( ξ ) ≍ - log ( 1 + | ξ | β ) for β ∈ ( 0 , 2 ] and Ψ ( ξ ) ≍ - ( log ( 1 + | ξ | β / 4 ) ) 2 for β ∈ ( 0 , 2 ) by taking ℓ ( r ) ≍ 1 and ℓ ( r ) ≍ log ( 1 + r β ) for r ≥ 1 respectively. In this article, we use the Calderón–Zygmund approach and function space theory for operators having slowly varying symbols.