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 Title
 An Lq(Lp)theory for spacetime nonlocal equations generated by Levy processes with low intensity of small jumps
 KIAS Author
 Park, Daehan
 Journal
 STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONSANALYSIS AND COMPUTATIONS, 2024
 Archive

 Abstract
 We investigate an Lq(Lp)regularity (1 < p, q < infinity) theory for spacetime nonlocal equations of the type partial derivative(alpha)(t) u = Lu + f. Here, partial derivative(alpha)(t) is the Caputo fractional derivative of order alpha is an element of(0, 1) and L is an integrodifferential operator Lu(x) = integral(Rd) (u(x)  u(x + y)  del u(x) center dot y1(vertical bar y vertical bar <= 1)) j(d) (vertical bar y vertical bar)dy which is the infinitesimal generator of an isotropic unimodal Levy 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 C1 (R/r)(delta 1) <= l(R)/l(r) <= C2 (R/r)(delta 2) for 1 <= r <= R < infinity, where 0 <= delta(1) <= delta(2) < 2. Hence, l can be slowly varying at infinity. Our result covers L whose Fourier multiplier Psi(xi) satisfies Psi(xi) asymptotic to  log (1 + vertical bar xi vertical bar(beta)) for beta is an element of (0, 2] and Psi (xi) asymptotic to (log(1+vertical bar xi vertical bar(beta/4)))(2) for beta is an element of (0, 2) by taking l(r) asymptotic to 1 and l(r) asymptotic to log (1 + r(beta)) for r >= 1 respectively. In this article, we use the CalderonZygmund approach and function space theory for operators having slowly varying symbols.