在变指数$p(x)$的标准假设（对数和衰减条件）下，利用Poisson半群$P_t$的收敛速率给出了变指数Bessel势空间$\mathfrak B^α[L^{p(\cdot)}(\mathbb R^n)]$的一个刻画.证明了在$L^{p(\cdot)}(\rn)$空间中Riesz分数阶导数$\mathbb{D}^\al f$的存在性等价于极限$\frac{1}{\ve^\al}(I-P_\ve)^\al f$的存在性.在预极限情形$\sup_x p(x)<\frac{n}{\al}$中，我们证明了贝塞尔势空间的特征满足条件$\|(I-P_\ve)^\al f\|_{p(\cdot)}\leqq C \ve^\al$
英文摘要：
Under the standard assumptions on the variable exponent $p(x)$ (log- and decay conditions), we give a characterization of the variable exponent Bessel potential space $\mathfrak B^α[L^{p(\cdot)}(\mathbb R^n)]$ in terms of the rate of convergence of the Poisson semigroup $P_t$. We show that the existence of the Riesz fractional derivative $\mathbb{D}^\al f$ in the space $L^{p(\cdot)}(\rn)$ is equivalent to the existence of the limit $\frac{1}{\ve^\al}(I-P_\ve)^\al f$. In the pre-limiting case $\sup_x p(x)<\frac{n}{\al}$ we show that the Bessel potential space is characterized by the condition $\|(I-P_\ve)^\al f\|_{p(\cdot)}\leqq C \ve^\al$
In this paper we analyze the convergence of the following type of series \begin{equation*} T_N f(x)=\sum_{j=N_1}^{N_2} v_j\Big(\mathcal{P}_{a_{j+1}} f(x)-\mathcal{P}_{a_{j}} f(x)\Big),\quad x\in \mathbb R_+, \end{equation*} where $\{\mathcal{P}_t \}_{t>0}$ is the Poisson semigroup of the Bessel operator $\displaystyle Δ_λ:=-{d^2\over dx^2}-{2λ\over x}{d\over dx}$ with $λ$ being a positive constant, $N=(N_1, N_2)\in \mathbb Z^2$ with $N_1<N_2,$ $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence. {Our analysis will consist in the …
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
Construction of Reproducing Kernel Hilbert Spaces Associated with the Laplace--Bessel Operator
Voronezh State University（沃罗涅什国立大学）；Belgorod State National Research University (BelGU)（别尔哥罗德国立研究大学）；Kadyrov Chechen State University（卡德罗夫车臣国立大学）；National Research University Higher School of Economics（国立高等经济大学）
In this paper we characterize the discrete Hölder spaces by means of the heat and Poisson semigroups associated to the discrete Laplacian. These characterizations allow us to get regularity properties of fractional powers of the discrete Laplacian and the Bessel potentials along these spaces and also in the discrete Zygmund spaces in a more direct way than using the pointwise definition of the spaces. To obtain our results, it has been crucial to get boundedness properties of the heat and Poisson kernels and their derivatives in both space and time variables. We believe that these estimates ar…
For $1<p\leq q$ we show that the Poisson semigroup $e^{-t\sqrt{-Δ}}$ on the $n$-sphere is hypercontractive from $L^{p}$ to $L^{q}$ in dimensions $n \leq 3$ if and only if $e^{-t\sqrt{n}} \leq \sqrt{\frac{p-1}{q-1}}$. We also show that the equivalence fails in large dimensions.
Non-Archimedean Pseudo-Differential Operators With Bessel Potentials
In this article, we study a class of non-archimedean pseudo-differential operators associated via Fourier transform to the Bessel potentials. These operators (which we will denote as $J^{α},$ $α>n$ ) are of the form (J^{α})(x)= \mathcal{F}_{ξ\rightarrow x}^{-1} \left [ ( \max \{1,||ξ||_{p}\})^{-α} \widehat{φ} (ξ) \right ] , \text{ } φ \in \mathcal{D} \mathbb{Q}_{p}^{n} ), \text{ } x \in \mathbb{Q}_{p}^{n} . We show that the fundamental solution $Z(x,t)$ of the $p-$ adic heat equation naturally associated to these operators satisfies $Z(x,t)<= 0,x \in \mathbb{Q} _{p}^{n},t>0. So this equation d…
We use Bochner's subordination technique to obtain caloric smoothing estimates in Besov- and Triebel--Lizorkin spaces. Our new estimates extend known smoothing results for the Gauß--Weierstraß, Cauchy--Poisson and higher-order generalized Gauß--Weierstraß semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched.
We start by identifying a class of pseudo-differential operators, generated by the set of continuous negative definite functions, that are in the weak similarity (WS) orbit of the self-adjoint log-Bessel operator on the Euclidean space. These WS relations turn out to be useful to first characterize a core for each operator in this class, which enables us to show that they generate a class, denoted by $\mathscr{P}$, of non-self-adjoint $\mathcal{C}_0$-contraction positive semigroups. Up to a homeomorphism, $\mathscr{P}$ includes, as fundamental objects in probability theory, the family of self-…
User's guide to the fractional Laplacian and the method of semigroups
\textit{method of semigroups}是一个统一的，广泛适用的，一般的技术，制定和分析的基本方面的分数权力的运营商$L$和他们的正则性在相关的功能空间。该方法是由作者和José L. 2009年，Torrea（arXiv：0910.2569v1）。本章的目的是展示该方法如何在分数拉普拉斯$L^s=(-Δ)^s$、$0<s<1$的特殊情况下工作。本文的出发点是分数拉普拉斯算子的半群公式。从这里，经典的热核允许我们获得逐点公式为$(-Δ)^su(x)$。关键优势之一是我们的技术依赖于热核的使用，这允许在傅里叶变换不是最合适工具的环境中应用。此外，它提供了明确的常数是关键证明，在最小条件下$u$，逐点限制的有效性$$\lim_{s\to1^-}(-Δ)^su(x)=-Δu(x)\quad\hbox{and}\quad\lim_{s\to0^+}(-Δ)^su(x)=u(x).$$的公式的解决方案的泊松问题$(-Δ)^su=f$是通过半群的方法作为逆的分数拉普拉斯$u(x)=(-Δ)^{-s}f(x)$（基本解决方案）。然后，我们提出的Caffarelli-西尔维斯特扩展问题，其明确的解决方案是由半群公式，首先发现了作者和Torrea。利用延拓技巧，得到了分数阶调和函数的一个内部Harnack不等式和导数估计.用半群的方法证明了经典的Hölder估计和Schauder估…
