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Questions tagged [function-spaces]

2 votes
0 answers
83 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 ...
jeffreygorwinkle's user avatar
6 votes
1 answer
203 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-...
Some random guy's user avatar
0 votes
1 answer
284 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 ...
feltshire's user avatar
  • 111
2 votes
1 answer
176 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 ...
Guillermo García Sáez's user avatar
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, ...
Goulifet's user avatar
  • 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 $\mathcal F:Vect(M)\rightarrow Ban$...
Timmathy's user avatar
  • 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_{\...
Liding Yao's user avatar
  • 1,481
4 votes
0 answers
85 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 \...
Guillermo García Sáez's user avatar
0 votes
0 answers
86 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 $\limsup \frac{f}{g} < \infty$,...
Leon Kim's user avatar
  • 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=\...
Guillermo García Sáez's user avatar
1 vote
0 answers
218 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 ...
Guillermo García Sáez's user avatar
2 votes
1 answer
175 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 ...
Guillermo García Sáez's user avatar
8 votes
0 answers
300 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 ...
CBBAM's user avatar
  • 873
2 votes
1 answer
232 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 ...
Anacardium's user avatar
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 $\...
Anacardium's user avatar
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}...
User091099's user avatar
0 votes
0 answers
161 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 $\alpha$ ...
Liding Yao's user avatar
  • 1,481
4 votes
1 answer
158 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_{...
Liding Yao's user avatar
  • 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^\...
Liding Yao's user avatar
  • 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, ...
ash's user avatar
  • 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 ...
CBBAM's user avatar
  • 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 ...
user90189's user avatar
  • 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://...
fast_and_fourier's user avatar
4 votes
1 answer
341 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)...
Hiro Shiba's user avatar
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-...
shuhalo's user avatar
  • 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 ...
Emily's user avatar
  • 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 \...
shuhalo's user avatar
  • 5,525
12 votes
1 answer
539 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 ...
Tobias Fritz's user avatar
  • 6,876
6 votes
0 answers
260 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), ...
Hannes's user avatar
  • 2,790
1 vote
0 answers
127 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 (...
Mir Aaliya's user avatar
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 ...
user92646's user avatar
  • 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})} &...
Seven9's user avatar
  • 565
3 votes
1 answer
504 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 ...
AB_IM's user avatar
  • 4,942
1 vote
1 answer
716 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 ...
Ho Man-Ho's user avatar
  • 1,263
2 votes
1 answer
242 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 ...
SetValued_Michael's user avatar
0 votes
0 answers
221 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 $L_\infty(0,1)$?...
user92646's user avatar
  • 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:...
Daniella Dannell's user avatar
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 ...
epitaph's user avatar
  • 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 ...
Alan's user avatar
  • 71
2 votes
0 answers
78 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 $C(X)$.
Mir Aaliya's user avatar
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.
Mir Aaliya's user avatar
3 votes
0 answers
397 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) \|...
shuhalo's user avatar
  • 5,525
2 votes
1 answer
265 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 ...
mw19930312's user avatar
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 "...
Chris Wendl's user avatar
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} &...
shuhalo's user avatar
  • 5,525
1 vote
1 answer
194 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, \...
A. Bailleul's user avatar
  • 1,352
6 votes
2 answers
812 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. ...
Glen M Wilson's user avatar
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 $C(\mathbb{R}^n;\mathbb{R}^d)$? $X$ ...
AB_IM's user avatar
  • 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 ...
Thomas Hank Clayton's user avatar
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 ...
Elizeu França's user avatar