arXiv subjects

Michal Johanis

Publications and source records attributed to Michal Johanis.

6 recordsLinked to original sources

arXiv2025

On Whitney-type extension theorems for $C^{1,+}$, $C^2$, $C^{2,+}$, and $C^3$-smooth mappings between Banach spaces

In 1973 J. C. Wells showed that a variant of the Whitney extension theorem holds for $C^{1,1}$-smooth real-valued functions on Hilbert spaces. In 2021 D. Azagra and C. Mudarra generalised this result to $C^{1,\omega}$-smooth functions on certain super-reflexive spaces. We show that while the vector-valued version of these results do hold in some rare cases (when the target space is an injective Banach space, e.g. $\ell_\infty$), it does not hold for mappings from infinite-dimensional spaces into "somewhat euclidean" spaces (e.g. infinite-dimensional spaces of a non-trivial type), and neither does the $C^2$-smooth variant. Further, we prove negative results concerning the real-valued $C^{2,+}$, $C^{2,\omega}$, and $C^3$-smooth versions generalising older results of J. C. Wells.

math.FA↗

arXiv2024

On $C^1$ Whitney extension theorem in Banach spaces

Our note is a complement to recent articles \cite{JS1} (2011) and \cite{JS2} (2013) by M. Jim\'enez-Sevilla and L. S\'anchez-Gonz\'alez which generalise (the basic statement of) the classical Whitney extension theorem for $C^1$-smooth real functions on $\mathbb R^n$ to the case of real functions on $X$ (\cite{JS1}) and to the case of mappings from $X$ to $Y$ (\cite{JS2}) for some Banach spaces $X$ and $Y$. Since the proof from \cite{JS2} contains a serious flaw, we supply a different more transparent detailed proof under (probably) slightly stronger assumptions on $X$ and $Y$. Our proof gives also extensions results from special sets (e.g. Lipschitz submanifolds or closed convex bodies) under substantially weaker assumptions on $X$ and $Y$. Further, we observe that the mapping $F\in C^1(X;Y)$ which extends $f$ given on a closed set $A\subset X$ can be, in some cases, $C^\infty$-smooth (or $C^k$-smooth with $k>1$) on $X\setminus A$. Of course, also this improved result is weaker than Whitney's result (for $X=\mathbb R^n$, $Y=\mathbb R$) which asserts that $F$ is even analytic on $X\setminus A$. Further, following another Whitney's article and using the above results, we prove results on extensions of $C^1$-smooth mappings from open ("weakly") quasiconvex subsets of $X$. Following the above mentioned articles we also consider the question concerning the Lipschitz constant of $F$ if $f$ is a Lipschitz mapping.

math.FA↗

arXiv2024

Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(\Ga)$

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into $c_0(\Ga)$ and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces $L_p(\mu)$, $1\le p\le2$) solving open problems posed by C.~Avart, P.~Komjath, and V.~Roedl and by G.~Godefroy, G.~Lancien, and V.~Zizler. In particular, we show that $X=L_p(\mu)$, $1\le p<\infty$, bi-Lipschitz embeds into $c_0(\Ga)$ if and only if $\dens X<\om_\om$.

math.FA↗

arXiv2023

On Whitney-type extension theorems on Banach spaces for $C^{1,\omega}$, $C^{1,+}$, $C_{\mathrm{loc}}^{1,+}$, and $C_{\mathrm B}^{1,+}$-smooth functions

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021). We show how older results on semiconvex functions with modulus $\omega$ easily imply extension theorems for $C^{1,\omega}$-smooth functions on super-reflexive Banach spaces which are versions of some theorems of Azagra and Mudarra. We present also some new interesting consequences which are not mentioned in their article, in particular extensions of $C^{1,\omega}$-smooth functions from open quasiconvex sets. They proved also an extension theorem for $C_{\mathrm B}^{1,+}$-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces. Our version of this theorem and new extension results for $C^{1,+}$ and $C_{\mathrm{loc}}^{1,+}$-smooth functions (i.e., functions with uniformly, resp. locally uniformly continuous derivative), all of which are proved on arbitrary super-reflexive Banach spaces, are further main contributions of our paper. Some of our proofs use main ideas of the article by D. Azagra and C. Mudarra, but all are formally completely independent on their article.

math.FA↗

arXiv2016

Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.

math.FA↗

arXiv2009

On Peano's theorem in Banach spaces

We show that if $X$ is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping $f\colon X\to X$ such that the autonomous differential equation $x'=f(x)$ has no solution at any point.

math.CA↗