Using a generalized spherical mean operator, we obtain a generalization of Titchmarsh's theorem for the Dunkl transform for functions satisfying the ('; p)-Dunkl Lipschitz condition in the space Lp(Rd;wl(x)dx), 1 < p 6 2, where wl is a weight function invariant under the action of an associated re ection group.

Let Ω be a strongly Lipschitz domain of Rn. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the non-tangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with...

Remark 1. The Lipschitz condition (4) is called global because it holds for all x, y ∈ R with the same constant L. Klebaner Theorem 5.4 assumes only a local Lipschitz condition, where the Lipschitz constant may depend on the size of x, y. We use a global condition in order to make the presentation simpler. Later on we will also assume a Lipschitz condition with respect to t, see (21). (There is...

(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extensio...

In this paper, using a generalized Jacobi-Dunkl translation operator, we obtain a generalization of Titchmarsh's theorem for the Dunkl transform for functions satisfying the Lipschitz Jacobi-Dunkl condition in the space Lp.

Using a generalized spherical mean operator, we obtain the generalizationof Titchmarsh's theorem for the Dunkl transform for functions satisfyingthe Lipschitz condition in L2(Rd;wk), where wk is a weight function invariantunder the action of an associated reection groups.

Using a Bessel generalized translation, we obtain an analog of Titchmarsh's theorem for the Bessel transform for functions satisfying the Lipschitz condition in the space $mathrm{L}_{p,alpha}(mathbb{R}_{+})$, where $alpha>-frac{1}{2}$ and $1

*Correspondence: [email protected] 1College of Information Sciences and Technology, Donghua University, Shanghai, China 2School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing, China Full list of author information is available at the end of the article Abstract In this paper, we study the Carathéodory approximate solution for a class of doubly perturbed st...

Canonical process is a Lipschitz continuous uncertain process with stationary and independent increments, and uncertain differential equation is a type of differential equations driven by canonical process. This paper presents some methods to solve linear uncertain differential equations, and proves an existence and uniqueness theorem of solution for uncertain differential equation under Lipsch...