arXiv:1907.11867v1 [math.PR] 27 Jul 2019 Maximal inequalities and exponential estimates for stochastic convolutions driven by L´ evy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations † Jiahui Zhu a , Zdzis law Brze´ zniak b , Wei Liu c, 1 a. School of Science, Zhejiang University of Technology, Hangzhou 310019, China b. Department of Mathematics, University of York, York YO10 5DD, UK c. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China Abstract. We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by L´ evy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itˆ o’s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with L´ evy noise is established. Keywords. Burkholder-Davis-Gundy inequality, maximal inequality, exponential estimate, stochastic con- volution, Itˆ o formula, martingale type r Banach space AMS Subject Classification. 60H15; 60J75; 35B65; 46B09 1. Introduction Over the past few decades, stochastic partial differential equations (SPDEs) have attracted considerable attention from researchers in a wide variety of fields, including biology, physics, engineering and finance etc. (cf. [11, 31] and the references therein). In the study of SPDEs, the Burkholder-Davis-Gundy (BDG) inequality and maximal inequality play vital roles in proving the existence, uniqueness, and regularity of solutions of SPDEs. There are quite a number of contributions on the study of BDG and maximal inequalities when the state space is a Hilbert space; see [10], [22], [23], [49] and [37]. However, many interesting problems in the theory of SPDEs whose natural settings in function spaces are not Hilbert spaces, but rather Banach spaces (e.g. some Sobolev spaces). Nevertheless, literature and research studies related to these inequalities on general Banach spaces are very limited and this is the motivation of our paper. The overall goal of this work is to investigate BDG inequalities and maximal regularities of stochastic convolutions driven by L´ evy processes in Banach spaces. We will derive, in Appendix B, two general versions of Itˆ o’s formula for L´ evy-type processes in Banach spaces, which are crucial for the proof of our inequalities. We will work in the martingale type r Banach spaces with 1 <r ≤ 2. This assumption is necessary for establishing a theory of stochastic integration in Banach spaces. Typical examples of such spaces are L p spaces with p ∈ [r, ∞) and Sobolev spaces. Now let us state our problem more explicitly. Let (E, |·| E ) be a separable Banach space of martingale type r with 1 <r ≤ 2 and (Ω, F , P) be a probability space with the filtration (F t ) t≥0 satisfying the usual hypotheses, and let (Z, Z ) be a measurable space. We first consider the following process u t =  t 0  Z ξ (s,z ) ˜ N (ds, dz ),t ∈ [0,T ],T> 0, † This work is supported by NSFC (11501509,11571147,11822106,11831014), NSF of Jiangsu Province (BK20160004), and the Qing Lan Project and PAPD of Jiangsu Higher Education Institutions. 1 Corresponding author: weiliu@jsnu.edu.cn