In this paper we analyze the convergence of the following type series$T_{N}f(x)=\sum _{j=N_1}^{N_2}{v}_j(e^{-a_{j+1}{\mathrm{\Delta }}^2}f(x)-e^{-a_j{\mathrm{\Delta }}^2}f(x)),x\in {\mathbb{R}}^n,$ where ${\{e^{-t{\mathrm{\Delta }}^2}\}}_{t>0}$ is the heat semigroup of the biharmonic operator ${\mathrm{\Delta }}^2$ with Δ being the classical laplacian, $N=(N_1,N_2)\in {\mathbb{Z}}^2$ with $N_1<N_2$, ${\{v_j\}}_{j\in \mathbb{Z}}$ is a bounded real sequences and ${\{a_j\}}_{j\in \mathbb{Z}}$ is a ρ-lacunary sequence of positive numbers, that is, $1<\rho \le a_{j+1}/a_j,\text{for all}j\in \mathbb{Z}$. Our analysis will consist in the boundedness, in $L^p({\mathbb{R}}^n)$ and in $BMO({\mathbb{R}}^n)$, of the operators $T_N$ and its maximal operator $T^{⁎}f(x)=\underset{N}{\sup }|T_{N}f(x)|$. The proofs of these results need the language of semigroups in an essential way.

在本文中，我们分析以下类型系列的收敛性${吨}_{N}F(X)=\sum _{j=N_1}^{N_2}{v}_j({\mathrm{电子}}^{-{\mathrm{一种}}_{j+1}{\mathrm{\Delta }}^2}F(X)-{\mathrm{电子}}^{-{\mathrm{一种}}_j{\mathrm{\Delta }}^2}F(X)),X\in {\mathbb{电阻}}^n,$ 在哪里 ${\{{\mathrm{电子}}^{-吨{\mathrm{\Delta }}^2}\}}_{吨>0}$ 是双调和算子的热半群 ${\mathrm{\Delta }}^2$ 其中 Δ 是经典的拉普拉斯算子， $N=(N_1,N_2)\in {\mathbb{Z}}^2$ 和 $N_1<N_2$, ${\{v_j\}}_{j\in \mathbb{Z}}$ 是一个有界实数序列并且 ${\{{\mathrm{一种}}_j\}}_{j\in \mathbb{Z}}$是一个ρ -空缺的正数序列，即$1<\rho \le {\mathrm{一种}}_{j+1}/{\mathrm{一种}}_j,\text{对所有人}j\in \mathbb{Z}$. 我们的分析将在于有界性，在${升}^{磷}({\mathbb{电阻}}^n)$ 并在 $乙米哦({\mathbb{电阻}}^n)$, 运营商 ${吨}_N$ 及其最大算子 ${吨}^{⁎}F(X)=\underset{N}{\mathrm{超}}|{吨}_{N}F(X)|$. 这些结果的证明在本质上需要半群语言。