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- Title
- An Lq(Lp)-theory for space-time non-local equations generated by Levy processes with low intensity of small jumps
- KIAS Author
- Park, Daehan
- Journal
- STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS, 2024
- Archive
-
- Abstract
- We investigate an L-q(L-p)-regularity (1 < p, q < infinity) theory for space-time 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 integro-differential 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 C-1 (R/r)(delta 1) <= l(R)/l(r) <= C-2 (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 Calderon-Zygmund approach and function space theory for operators having slowly varying symbols.