The aim of the paper is two-fold. First, we investigate the ψ-Bessel potential spaces on ${ℝ}_{0+}^{n+1}$ and study some of their properties. Secondly, we consider the fractional powers of an operator of the form $-A_{\pm }=-\psi \left(D_{x^{\text{'}}}\right)\pm \partial /\left(\partial x_{n+1}\right)$, $(x^{\text{'}},x_{n+1})\in {ℝ}_{0+}^{n+1}$, where $\psi \left(D_{x^{\text{'}}}\right)$ is an operator with real continuous negative definite symbol ψ: ℝⁿ → ℝ. We define the domain of the operator $-{(-A_{\pm })}^{\alpha }$ and prove that with this domain it generates an $L_p$-sub-Markovian semigroup.

Let be a locally compact Hausdorff space. Let $A_i$, i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes $X_i=X_i\left(t\right),t\ge 0$ and let ${\alpha }_i$, i = 0,...,N, be non-negative continuous functions on with ${\sum }_{i=0}^N{\alpha }_i=1$. Assume that the closure A of ${\sum }_{k=0}^N{\alpha }_k{A}_k$ defined on ${\bigcap }_{i=0}^N\left(A_i\right)$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). A natural interpretation of a related Feller process X = X(t), t ≥ 0 is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ , with probability ${\alpha }_i\left(p\right)$, the process...

We study asymptotic behavior of $C_0$-semigroups T(t), t ≥ 0, such that ∥T(t)∥ ≤ α(t), where α(t) is a nonquasianalytic weight function. In particular, we show that if σ(A) ∩ iℝ is countable and Pσ(A*) ∩ iℝ is empty, then $li{m}_{t\to \infty }1/\alpha \left(t\right)\parallel T\left(t\right)x\parallel =0$, ∀x ∈ X. If, moreover, f is a function in $L_{\alpha }^1\left({ℝ}_{+}\right)$ which is of spectral synthesis in a corresponding algebra $L_{{\alpha }_1}^1\left(ℝ\right)$ with respect to (iσ(A)) ∩ ℝ, then $li{m}_{t\to \infty }1/\alpha \left(t\right)\parallel T\left(t\right)f̂\left(T\right)\parallel =0$, where $f̂\left(T\right)={ʃ}_0^{\infty }f\left(t\right)T\left(t\right)dt$. Analogous results are obtained also for iterates of a single operator. The results are extensions of earlier results of Katznelson-Tzafriri,...

Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Hölder spaces illustrates the abstract result.

This paper is devoted to the investigation of the abstract semilinear initial value problem du/dt + A(t)u = f(t,u), u(0) = u₀, in the "parabolic" case.

The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the...

Let $ℒ$ be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure $\gamma$ on ${ℝ}^d.$ We prove a sharp estimate of the operator norm of the imaginary powers of $ℒ$ on $L^p\left(\gamma \right),$$1\<p\<\infty ...$

We prove a characterisation of sets with finite perimeter and $BV$ functions in terms of the short time behaviour of the heat semigroup in ${\mathbf{R}}^N$. For sets with smooth boundary a more precise result is shown.