Vector-valued Littlewood-Paley-Stein theory for semigroups

We develop a generalized Littlewood-Paley theory for semigroups acting on $L^p$-spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein $g$-function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. We show that in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on ${\mathbb R}^n$, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder\'on-Zygmund singular integral operators.Comment: To appear in Adv. Mat

Non-local fractional derivatives. Discrete and continuous

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough, these approximation procedures give a measure of the order of approximation. These results also allows us to prove the coincidence, for good enough functions, of the Marchaud and Gr\"unwald-Letnikov derivatives in every point and the speed of convergence to the Gr\"unwald-Letnikov derivative. The fractional discrete derivative will be also described as a Neumann-Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).

Lipschitz spaces adapted to Schrödinger operators and regularity properties

Consider the Schrödinger operator L= - Δ + V in Rn, n≥ 3 , where V is a nonnegative potential satisfying a reverse Hölder condition of the type (1|B|∫BV(y)qdy)1/q ≤C|B|∫BV(y)dy, for some q > n/2. We define ΛαL, 0 0‖f(·+z) + f(·-z)-2f(·)‖∞|z|α 0 , we denote by Λ Wα/2 the set of functions f which satisfy ‖ρ(·)-αf(·)‖∞ 0. We prove that for 0 < α ≤ 2 - n/q, ΛαL = Λ Wα/2. As application, we obtain regularity properties of fractional powers (positive and negative) of the operator L, Schrödinger Riesz transforms, Bessel potentials and multipliers of Laplace transforms type. The proofs of these results need in an essential way the language of semigroups. Parallel results are obtained for the classes defined through the Poisson semigroup, Py f = e-y √LfMarta De León-Contreras was partially supported by grant EPSRC Research Grant EP/S029486/1. José L. Torrea was partially supported by Grant PGC2018-099124-B-I00 (MINECO/FEDER

Boundedness of differential transforms for one-sided fractional poisson-type operator sequence

“This is a post-peer-review, pre-copyedit version of an article published in Journal of Geometric Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s12220-019-00251-x”The first author was supported by National Natural Science Foundation of China (Grant Nos. 11971431, 11401525), the Natural Science Foundation of Zhejiang Province (Grant No. LY18A010006), the first Class Discipline of Zhejiang - A (Zhejiang Gongshang University- Statistics) and the State Scholarship Fund (No. 201808330097). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11671308, 11431011) and the independent research project of Wuhan University (Grant No. 2042017kf0209). The third author was supported by grant MTM2015-66157-C2-1-P (MINECO/FEDER) from Government of Spai

Non-local fractional derivatives. Discrete and continuous

We prove maximum and comparison principles for the discrete fractional derivatives in the integers. Regularity results when the space is a mesh of length h, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough (Hölder continuous), these approximation procedures give a measure of the order of approximation. These results also allow us to prove the coincidence, for Hölder continuous functions, of the Marchaud and Grünwald–Letnikov derivatives in every point and the speed of convergence to the Grünwald–Letnikov derivative. The discrete fractional derivative will be also described as a Neumann–Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces lp(Z)