In the first note we show for a strongly continuous family of operators ${\left(T\left(t\right)\right)}_{t\ge 0}$ that if every orbit $t\mapsto T\left(t\right)x$ is differentiable for $t>t_x$, then all orbits are differentiable for $t>t_0$ with $t_0$ independent of $x$. In the second note we give an example of an eventually differentiable semigroup which is not differentiable on the same interval in the operator norm topology.

R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on $L^2$ maps $L^{\infty }$ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by $C\left|R{|}^s\right|R{|}^{-s}$, for some $s\>0$. We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided $s\>s_0\left(E\right)$. We also show that the Calderon-Coifman bicommutators,...

In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.

We establish two fixed point theorems for certain mappings of contractive type.

We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in ${ℝ}^n$. Also, for symmetric second-order ordinary differential operators we show that ${lim\; sup}_{t\to c}{\left(p{q}^{\text{'}}\right)}^{\text{'}}/q^2=\theta <2$ where $c$ is a singular point guarantees separation of $-{\left(p{y}^{\text{'}}\right)}^{\text{'}}+qy$ on its minimal domain and extend this criterion to the partial differential setting. As a particular example it is shown that $-\Delta y+qy$ is separated on its minimal domain if $q$ is superharmonic. For $n=1$ the criterion...

We give some explicit values of the constants $C_1$ and $C_2$ in the inequality $C_1/sin\left(\frac{\pi }{p}\right)\le {\left|P\right|}_p\le C_2/sin\left(\frac{\pi }{p}\right)$ where ${\left|P\right|}_p$ denotes the norm of the Bergman projection on the $L^p$ space.

New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator Ms [resp. Is], 0 ≤ s &lt; n, [resp. 0 &lt; s &lt; n] sends the weighted Lebesgue space Lp(v(x)dx) into Lp(u(x)dx), 1 &lt; p &lt; ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.

Let V be a two-dimensional real symmetric space with unit ball having 8n extreme points. Let λ(V) denote the absolute projection constant of V. We show that $\lambda \left(V\right)\le \lambda \left(V_n\right)$ where $V_n$ is the space whose ball is a regular 8n-polygon. Also we reprove a result of [1] and [5] which states that $4/\pi =\lambda \left(l{₂}^{\left(2\right)}\right)\ge \lambda \left(V\right)$ for any two-dimensional real symmetric space V.

In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.