The convergence of Rothe’s method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe’s approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.

We consider the equation $$-r\left(x\right)y^{\text{'}}\left(x\right)+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ$$ where $f\in L_p\left(ℝ\right)$, $p\in [1,\infty ]$ ($L_{\infty }\left(ℝ\right):=C\left(ℝ\right)$) and $$0<r\in C^{}\left(ℝ\right),0\le q\in L_1^{}\left(ℝ\right).$$ We obtain minimal requirements to the functions $r$ and $q$, in addition to (), under which equation () is correctly solvable in $L_p\left(ℝ\right)$, $p\in [1,\infty ]$.

The purpose of this note is twofold. First it is a corrigenda of our paper [RV1]. And secondly we make some remarks concerning the interpolation properties of Morrey spaces.

We are concerned with imbeddings of general spaces of Besov and Lizorkin-Triebel type with dominating mixed derivatives in the first critical case. We employ multivariate exponential Orlicz and Lorentz-Orlicz spaces as targets. We study basic properties of the target spaces, in particular, we compare them with usual exponential spaces, showing that in this case the multivariate clones are in fact better adapted to the character of smoothness of the imbedded spaces. Then we prove sharp limiting imbedding...

Let $\Theta :={\theta }_I^e:e\in E,I\in D$ be a decomposition system for $L₂\left({ℝ}^d\right)$ indexed over D, the set of dyadic cubes in ${ℝ}^d$, and a finite set E, and let $\Theta ̃:=\Theta {̃}_I^e:e\in E,I\in D$ be the corresponding dual functionals. That is, for every $f\in L₂\left({ℝ}^d\right)$, $f={\sum }_{e\in E}{\sum }_{I\in D}⟨f,\Theta {̃}_I^e⟩{\theta }_I^e$. We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨f,\Theta {̃}_I^e⟩$, e ∈ E, I ∈ D. Typical examples of such decomposition systems...

There have been recent attempts to develop the theory of Sobolev spaces $W^{1,p}$ on metric spaces that do not admit any differentiable structure. We prove that certain definitions are equivalent. We also define the spaces in the limiting case $p=1$.

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\&lt;\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.