The Teodorescu Operator in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions,i.e.,null solutions to a first order vector valued rotation invariant differential operator (θ) ca...     

Hermitian generalization of the Rarita-Schwinger operators

We introduce two new linear differential operators which are invariant with respect to the unitary group SU（n）. They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

The waiting spectra of the sets described by the quantitative waiting time indicators

The waiting spectra of the sets consisting of pairs of sequences with prescribed quantitative waiting time indicators are determined. More precisely,let R(x,y) and R(x,y) be the lower and upper quantitative waiting time indicators of y by x respectively in the symbolic space Σm(integer m 2) and define the level sets Sα,β={(x,y)∈Σ2m:R(x,y)=α,R(x,y)=β},where 0αβ∞,it is shown that the sets Sα,βare all of Hausdorff dimension 2.Besides,some further extensions of this result are also made.     

Recurrence in β-expansion over formal Laurent series

In this paper, we study the quantitative recurrence and hitting sets of β-transformation T _β on the unit disk I of formal Laurent series field $$E_\phi:= \{x\in I: \|T_\beta^nx - x\| < \|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\}$$ and $$F_\phi:=\{x\in I: \|T_\beta^nx-x_0\|<\|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\},$$ where x ₀ is any fixed point in I and ${\phi}$ is any positive function defined on ${\mathbb{N}}$ with ${\phi(n)\to\infty}$ as n → ∞. We completely determine the Hausdorff dimensions of these sets: $$\dim_{\rm H} E_{\phi}=\dim_{\rm H}F_\phi=\frac{1}{1+\liminf\limits_{n\to\infty}\frac{\phi (n)}{n}}.$$      

Symmetry breaking bifurcation from solutions concentrating on the equator of \mathbb{S}^N

We are concerned with the elliptic problem $${\varepsilon ^2}{\Delta _{{S^n}}}u - u + {u^p} = 0{\text{ in }}{S^n},u > 0{\text{ in }}{S^n}$$ , where ${\Delta _{{S^n}}}$ is the Laplace-Beltrami operator on $\mathbb{S}^n : = \left\{ {x \in \mathbb{R}^{n + 1} ;\left\| x \right\| = 1} \right\}\left( {n \geqslant 3} \right)$ , and p ? 2. We construct a smooth branch C of solutions concentrating on the equator S ⁿ ∩ {x _{n+1 = 0}}. Using the Crandall-Rabinowitz bifurcation theorem, we show that C has infinitely many bifurcation points from which continua of nonradial solutions emanate. In applying the bifurcation theorem, we verify the transversality condition directly.