Spaces with corner singularities, locally modelled by cones with base spaces having conical singularities, belong to the hierarchy of (pseudo-) manifolds with piecewise smooth geometry. We consider a typical case of a manifold with corners, the so-called "edged spindle", and a natural algebra of pseudodifferential operators on it with special degeneracy in the symbols, the "corner algebra". There are three levels of principal symbols in the corner algebra, namely the interior,...

We define homogeneous classes of x-dependent anisotropic symbols ${Ṡ}_{\gamma ,\delta }^m\left(A\right)$ in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent symbols in...

We prove the following result: Let X be a real Hilbert space and let J: X → ℝ be a C¹ functional with a nonexpansive derivative. Then, for each r > 0, the following alternative holds: either J’ has a fixed point with norm less than r, or $su{p}_{\left|\right|x\left|\right|=r}J\left(x\right)=su{p}_{{\left|\right|u\left|\right|}_{L²\left(\right[0,1],X)}=r}{\int }_0^{1}J\left(u\left(t\right)\right)dt$.

In the present paper initial operators for a right invertible operator, which are induced by sequential shifts and have the property c(R) are constructed. An application to the Lagrange type interpolation problem is given. Moreover, an example with the Pommiez operator is studied.

In the paper [13] we proved a fixed point theorem for an operator $𝒜$, which satisfies a generalized Lipschitz condition with respect to a linear bounded operator $A$, that is: $$m(𝒜x-𝒜y)\prec Am(x-y).$$ The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator $A$.

We obtain modular convergence theorems in modular spaces for nets of operators of the form $\left(T_{w}f\right)\left(s\right)={\int }_H{K}_w(s-h_w\left(t\right),f\left(h_w\left(t\right)\right))d{\mu }_H\left(t\right)$, w > 0, s ∈ G, where G and H are topological groups and ${h_w}_{w>0}$ is a family of homeomorphisms $h_w:H\to h_w\left(H\right)\subset G.$ Such operators contain, in particular, a nonlinear version of the generalized sampling operators, which have many applications in the theory of signal processing.

Asymptotic convergence theorems for semigroups of nonnegative operators on a Banach lattice, on C(X) and on $L^p\left(X\right)$ (1 ≤ p ≤ ∞) are proved. The general results are applied to a class of semigroups generated by some differential equations.

New sufficient conditions for asymptotic stability of Markov operators are given. These criteria are applied to a class of Volterra type integral operators with advanced argument.

A new theorem on asymptotic stability and sweeping of substochastic semigroups is proved, and applied semigroups generated by birth-death processes.

We consider a periodic pseudo-differential operator on the real line, which is a lower-order perturbation of an elliptic operator with a homogeneous symbol and constant coefficients. It is proved that the density of states of such an operator admits a complete asymptotic expansion at large energies. A few first terms of this expansion are found in a closed form.