Questions tagged [function-spaces]
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73 questions
2 votes
0 answers
115 views
Is there a 'determinant' of a two-variable function when treated as a linear map?
A two variable real function $F(y,x)$, defined on $[c,d] \times [a,b]$, can be thought of as a linear map between functions of different domains by: $$ g(y) = \int^a_bF(y,x)f(x)dx $$ This has very ...
6 votes
1 answer
204 views
Composition and compactly generated spaces
Let $X$ and $Y$ and $Z$ be compactly generated Hausdorff spaces. Is it true that the composition map $Z^Y \times Y^X \rightarrow Z^X$ is continuous? Here the function spaces are given the compact-...
- 1,276
0 votes
1 answer
292 views
Reference Request: Besov spaces are compactly embedded in Hölder spaces
In the paper Wasserstein GANs are Minimax Optimal Distribution Estimators [1], the authors state within Lemma 3.3, p.12, that Besov spaces [2] of generalized smoothness (i.e., with the added parameter ...
- 111
2 votes
1 answer
177 views
A question about weighted spaces
Let $1<p<\infty$ and $w\in A_p$ a weight in the Muckenhoupt class. In [Lemma 2.2] Fröhlich, A. The Stokes Operator in Weighted -Spaces I: Weighted Estimates for the Stokes Resolvent Problem in a ...
2 votes
0 answers
98 views
Extreme points of the unit ball of the James space $J$ and the James tree space $JT$
Consider the $J$ sequence space and $JT$ the James tree space. Both are separable non-reflexive Banach spaces containing no isomorphic copies of $\ell_1$. Moreover, $JT$'s dual is non-separable, ...
- 2,602
3 votes
0 answers
69 views
Eliasons Manifoldmodel
In 1967 Eliasson published his Article about geometry of manifolds of maps, in which he proposes a so called manifold model. He states roughly that a section functor $\mathcal F:Vect(M)\rightarrow Ban$...
- 131
2 votes
0 answers
89 views
Tensor operator extension on BMO spaces
Let $T:L^p_{\mathrm{loc}}(\mathbb R)\to C_0(\mathbb R)$ be a bounded linear operator where $p<\infty$ is a suitable large number. Then we have boundedness of the tensor extension $I\otimes T:L^p_{\...
- 1,481
4 votes
0 answers
86 views
Generalized real method of interpolation for Orlicz spaces
Consider the Orlicz spaces $L^A(\mathbb{R}^n)$. Under suitable conditions for the $N$-function $A(x,t)$, namely $(Inc)_p$, $(Dec)_q,$, for $1<p<q<\infty$, it is known that $$L^p\cap L^q \...
0 votes
0 answers
87 views
Is there a characterization of growth rates that are "regularly behaved"?
Assume every function is eventually nonnegative. In other words, we are interested in growth rates for measuring time complexity and such. $f = O(g)$ is equivalent to $\limsup \frac{f}{g} < \infty$,...
- 189
2 votes
0 answers
131 views
Bessel spaces and Triebel Lizorkin
It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
1 vote
0 answers
222 views
Fractional Sobolev embedding
Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
2 votes
1 answer
177 views
Separability is an interpolation property
I'm trying to prove that certain space, which can be obtained as an interpolation space, is separable. The fact that is separable is well known but i want to simplify it via interpolation. I haven't ...
8 votes
0 answers
305 views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here. Let $C^k(\mathbb{R}^n$) be the ...
- 873
2 votes
1 answer
233 views
Show that $\|P(f\circ\varphi_{\lambda})-\widetilde{f}(\lambda)\|_p=\|P(f\circ\varphi_{\lambda}-\overline{P(\overline{f}\circ\varphi_{\lambda}}))\|_p.$
Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
- 71
2 votes
0 answers
55 views
Deck transformation group of the basic polynomial map on a $G$-space
Let $G \subseteq GL_d (\mathbb C)$ be a finite pseudoreflection group (see here and here) acting on a domain $\Omega \subseteq \mathbb C^d$ by the right action $\sigma \cdot z = \sigma^{-1} z$ where $\...
- 71
2 votes
0 answers
63 views
Dual of homogeneous Triebel-Lizorkin
Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with $$ [f]^{p}_{\dot{F}^{s}_{p,q}...
- 31
0 votes
0 answers
165 views
Characterization for the multipliers of Schwartz space
Is the following true? A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following: For every $\alpha$ ...
- 1,481
4 votes
1 answer
160 views
Is there any example of linear operator which is bounded on all Besov spaces but not on Triebel-Lizorkin spaces
Is there any linear operator $T:S'(\mathbb R^n)\to S'(\mathbb R^n)$ such that $T:B_{pq}^s(\mathbb R^n)\to B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but there exist a $F_{...
- 1,481
4 votes
0 answers
101 views
Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$
On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
- 1,481
1 vote
0 answers
144 views
Looking for examples of kernels with scalar Pick property but not the complete Pick property
I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy. A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...
- 203
3 votes
2 answers
1k views
What is the relationship between Hölder spaces and differentiability?
I'm porting this question over from MSE as it did not get any responses other than one comment on there. Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources ...
- 873
1 vote
0 answers
195 views
A generalization of polynomials in one variable
Let us consider the space of polynomials $P^N$ of degree $\le N$. If $f\in P^N$ vanishes in $>N$ points, then $f\equiv 0$, but for any $N$ points, or fewer, there exists $f\neq 0$ vanishing at ...
- 438
4 votes
0 answers
168 views
Why do we work on homogeneous Besov/Triebel-Lizorkin spaces?
This question is mainly for understanding the history behind homogeneous spaces. There is extensive literature on Besov and Triebel-Lizorkin spaces. For instance, see the standard textbook: https://...
4 votes
1 answer
344 views
Is $T$ totally bounded when $C_u(T)$ is separable?
I'm seeking help with a question regarding the space of bounded and uniformly continuous functions $C_u(T,d)$, where $(T,d)$ is a metric space. In this context, $C_u(T)$ is a closed subspace of $C_b(T)...
- 235
1 vote
1 answer
124 views
Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
The following startles me. Let $\Omega \subseteq \mathbb R^n$ and write $W^{\sigma,1}(\Omega)$ for the fractional Sobolev space with norm $$|u|_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-...
- 5,525
2 votes
0 answers
75 views
The graph topologies for powersets
Given a topological space $X$ and a metric space $(Y,d_Y)$, there are a number of topologies one may put on the space $\mathcal{C}(X,Y)$ of continuous functions from $X$ to $Y$. Perhaps the most ...
- 12.6k
10 votes
2 answers
1k views
What happens if we consider functions of bounded variation that are not in $L^1$?
A function $f \in L^1(\mathbb R^n)$ is said to be of bounded variation if there exists a constant $C \geq 0$ such that $$ \int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx \leq C \sup_{ x \in \...
- 5,525
12 votes
1 answer
544 views
Are algebras of smooth functions formally smooth?
Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$? If it helps, feel free to assume that $M$ is compact. (This is not a joke ...
- 6,876
6 votes
0 answers
261 views
Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain
I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...
- 2,790
1 vote
0 answers
128 views
What is t-equivalence in function spaces?
In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...
- 31
2 votes
0 answers
104 views
quasi-Banach function spaces are subspace of $L_p$
It is well-known that any Banach rearrangement-invariant function space $X$ on $[0,1]$ is a subset of $L_1[0,1]$, and I can find a reference that any quasi-Banach rearrangement-invariant function ...
- 663
3 votes
0 answers
91 views
Relationship between Hardy-Orlicz space and the corresponding Orlicz space
For $p \in [1, \infty]$ the Hardy space $H_p$ is defined as the space of all analytic functions $f$ on the open disk satisfying $$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...
- 565
3 votes
1 answer
511 views
Schauder basis of $L^1_{\mathrm{loc}}(\mathbb{R}^n,H)$
$\newcommand{\loc}{\mathrm{loc}}$Let $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n),\mu)$ denote the Euclidean space $\mathbb{R}^n$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ equipped with ...
- 4,942
1 vote
1 answer
726 views
About the normability of the space of continuous functions
Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
- 1,273
2 votes
1 answer
243 views
Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$
Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty. Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped ...
0 votes
0 answers
227 views
Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?
I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...
- 663
2 votes
0 answers
70 views
Determining a space of differentiability
I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by $$ P(t,s)f:...
4 votes
2 answers
2k views
Topologies on space of compactly supported continuous functions
Let $X$ be a locally compact Hausdorff space. As far as I understand, the space $C_c(X) = C_c(X; \mathbb{C})$ of compactly supported continuous complex-valued functions on $X$ is (most?) often ...
- 99
7 votes
0 answers
273 views
Has this Banach algebra been studied?
Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm ...
- 71
2 votes
0 answers
79 views
Where can I find literature regarding cardinal invariants of a function space $C(X, Y)$ endowed with the Uniform or Fine topology?
I am working on Function Spaces as a topological space. I want to get a sample paper which studies the cardinal invariants on the function space $C(X, Y)$ rather than on $C(X)$.
- 31
1 vote
1 answer
111 views
What is the source to find cardinal invariants for a function space C(X, Y), equipped with uniform or fine topology?
I would like to know about the technique to check the cardinality properties for the function space C(X, Y), where X is a tychonoff space and Y a metric space, equipped with uniform or fine topology.
- 11
3 votes
0 answers
402 views
Which domains have a Poincare-Wirtinger inequality? Which don't?
A Poincare-Wirtinger inequality holds over a domain $\Omega \subseteq \mathbb R^n$ with exponentnt $1 \leq p \leq \infty$ if there exists $C(p,\Omega) > 0$ such that $\| u - \operatorname{avg}(u) \|...
- 5,525
2 votes
1 answer
267 views
Can we define geodesic in the space of compactly supported functions?
From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
- 273
14 votes
0 answers
1k views
strong topologies on $C_c^\infty$
UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
- 486
2 votes
0 answers
399 views
Analogue of Lipschitz continuity of $W^{1,\infty}$ for Hölder continuity and Sobolev-Slobodeckij spaces
A function $u : U \rightarrow \mathbb R$ is an element of the Hölder space $C^{\alpha}(U)$ if $\sup\limits_{x \in U} |u(x)| < \infty$ $\sup\limits_{x,y \in U} \dfrac{|u(x) - u(y)|}{|x-y|^\alpha} &...
- 5,525
1 vote
1 answer
196 views
On $B^1$ and $B^2$ almost-periodic functions
The Besicovitch class of $B^p$ almost-periodic functions is defined as the closure of the set of trigonometric polynomials (of the form $t \mapsto \sum_{n=1}^N a_n e^{i \lambda_n t}$ with $\lambda_1, \...
- 1,352
6 votes
2 answers
819 views
The set of embeddings is open in the strong Whitney topology
In Hirsch's book "Differential Topology," he claims in Chapter 2, Theorem 1.4 that the set of $C^1$-embeddings is open in the strong Whitney topology $C^1(M, N)$ where $M$ and $N$ are $C^1$ manifolds. ...
- 1,402
2 votes
1 answer
388 views
Topological spaces containing paths
Let $C(\mathbb{R}^n;\mathbb{R}^d)$ be the space of continuous functions with the uniform-convergence on compacts topology. What are function spaces $X$ building on $C(\mathbb{R}^n;\mathbb{R}^d)$? $X$ ...
- 4,942
1 vote
0 answers
124 views
Which set of functions/measures has range $\mathrm{L}^\infty$ under Fourier transformation
I have a question concerning the Fourier transformation. What I know is that $\mathrm{L}^{\infty}=\{\hat{u}:\ u\in Y\}$ for some space $Y$. Now, I want to specify the space $Y$. The question is, is ...
1 vote
0 answers
268 views
Uniform convergence over compacts subsets implies existence of a uiform convergente subsequence?
Let $H$ the group of all homeomorphisms of a locally compact second countable and totally bounded metric space $X$ onto itself, under the compact-open topology ($X$ is totally bounded if every ...
- 375