Let $L=-\Delta +V$ be a Schrödinger operator in ${\mathbb R}^n$ with $n\geq 3$, where $\Delta $ is the Laplace operator denoted by $\Delta =\sum ^{n}_{i=1}({\partial ^{2}}/{\partial x_{i}^{2}})$ and the nonnegative potential V belongs to the reverse Hölder class $(RH)_{q}$ with $q>n/2$. For $\alpha \in (0,1)$, we define the operator

$$ \begin{align*} T_N^{L^{\alpha}} f(x) =\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x)) \quad \mbox{for all }x\in \mathbb R^n, \end{align*} $$

where $\{e^{-tL^\alpha } \}_{t>0}$ is the fractional heat semigroup of the operator L, $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequence and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence.

We investigate the boundedness of the operator $T_N^{L^{\alpha }}$ and the related maximal operator $T^*_{L^{\alpha }}f(x):=\sup _N \vert T_N^{L^{\alpha }} f(x)\vert $ on the spaces $L^{p}(\mathbb {R}^{n})$ and $BMO_{L}(\mathbb {R}^{n})$, respectively. As extensions of $L^{p}(\mathbb {R}^{n})$, the boundedness of the operators $T_N^{L^{\alpha }}$ and $T^*_{L^{\alpha }}$ on the Morrey space $L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ and the weak Morrey space $WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ has also been proved.

We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

We characterize bounded mean oscillation in terms of the boundedness of commutators of various bilinear singular integral operators with pointwise multiplication. In particular, we study commutators of a wide class of bilinear operators of convolution type, including bilinear Calderón–Zygmund operators and the bilinear fractional integral operators.

In this paper, we study the ${{L}^{p}}$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on ${{L}^{p}}$ provided that their kernels satisfy a size condition much weaker than that for the classical Calderón–Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

We characterize the pairs of weights (u, v) for which the maximal operator is of weak and restricted weak type (p, p) with respect to u(x)dx and v(x)dx. As a consequence we obtain analogous results for We apply the results to the study of the Cesàro-α convergence of singular integrals.

In this paper we introduce numerical schemes for aone-dimensional kinetic model of the Boltzmann equation withdissipative collisions and variable coefficient of restitution. Inparticular, we study the numerical passage of the Boltzmannequation with singular kernel to nonlinear friction equations inthe so-called quasi elastic limit. To this aim we introduce aFourier spectral method for the Boltzmann equation [CITE]and show that the kernel modes that define the spectral methodhave the correct quasi elastic limit providing a consistentspectral method for the limiting nonlinear friction equation.