Under certain conditions, an element of a tensor product space can be identified with a compact operator and the singular value decomposition (SVD) applies to the latter. These conditions are not fulfilled in Sobolev spaces. In the previous part of this work (part I) [Z. Anal. Anwend. 39 (2020), 349–369], we introduced some preliminary notions in the theory of tensor product spaces. We analyzed low-rank

We prove the global solutions to the incompressible Hall-magnetohydrodynamics (Hall-MHD) equations with small or some large initial data in $\mathbb R^n (n \geq 3)$. Especially, Fujita–Kato type initial data as the incompressible Navier–Stokes equations are allowed. We also study the large time behavior of the solutions and obtain an optimal decay rate in the general Besov spaces. Different from all

Stimulated by the category theorems of Eisner and Serény in the setting of unitary and isometric $C_0$-semigroups on separable Hilbert spaces, we prove category theorems for Schrödinger semigroups. Speci cally, we show that, to a given class of Schrödinger semigroups, Baire generically the semigroups are strongly stable but not exponentially stable. We also present a typical spectral property of the

In this paper, by using the idea of linearizing maximal operators originated by Charles Fefferman and the $TT*$ method of Stein–Wainger, we establish a weighted inequality for vector valued maximal Carleson type operators with singular kernels proposed by Andersen and John on the weighted Lorentz spaces with vector-valued functions.

In the present note, we examine the behavior of some homothecy-invariant differentiation basis of rectangles in the plane satisfying the following requirement: for a given rectangle to belong to the basis, the ratio of the largest of its side-lengths by the smallest one (which one calls its shape) has to be a fixed real number depending on the angle between its longest side and the horizontal line

We consider the nonlinear eigenvalue problem $$Lx + \varepsilon N(x) = \lambda Cx, \quad \|x\|=1,$$ where $\varepsilon,\lambda$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and $N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $\Sigma$ of the solutions $(x,\varepsilon

In this paper we build up a criteria for fractional Orlicz–Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.

Let $X = [0, 1]^n, n \geq 1$. We show that the typical (in the sense of Baire category) compact subset of $X$ is not only a zero dimensional Cantor space but it satisfies the property of being strongly microscopic, which is stronger than dimension zero.

We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{*}_{s}-2}u \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $1 < p < q < \frac{N}{s}$, $\q=\frac{Nq}{N-sq}$, $\lambda > 0$ is a parameter, $h$ is a nontrivial bounded perturbation and $f$ is a

We present a partial Hölder regularity result for the gradient of solutions to quasimonotone systems: $$\mathrm div \:\mathbf A(\cdot,D\mathbf u) = \mathbf B(\cdot, D\mathbf u) \quad \text{in } \Omega,$$ on bounded domains in the weak sense. Here certain continuity, uniformly strictly quasimonotonicity, growth conditions are imposed on the coefficients, including an asymptotic Uhlenbeck behaviour close

A well known result from functional analysis states that any compact operator between Hilbert spaces admits a singular value decomposition (SVD). This decomposition is a powerful tool that is the workhorse of many methods both in mathematics and applied fields. A prominent application in recent years is the approximation of high-dimensional functions in a low-rank format. This is based on the fact