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Results tagged with fa.functional-analysis
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user 272040
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
1
vote
Continuity upgrade for nonlinear maps
Edit: I have rewritten my answer after learning more details
I have been wondering for a while whether there is an interesting answer concerning analytic functions. I have looked at the papers suggest …
- 1,075
14
votes
What standard Banach space is isomorphic to the completion of this different normed structur...
In what follows I will show that the closure of $\ell^1$ under the norm $|x|=\int_1^2|x|_pdp$ is nothing but the Orlicz space $L_\Phi$, where $\Phi$ is the function
$$\Phi(t)=\frac{t^2-|t|}{\ln|t|}$$
…
- 1,075
5
votes
1
answer
539
views
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ i...
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its elements have …
- 1,075
7
votes
Accepted
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ i...
Edit. For the sake of improving the quality of the post, I modified the proof to make it work for all $M>n$ after Prof. Tao’s comments (the previous version was admittedly way too loose).
In the end I …
- 1,075
4
votes
0
answers
137
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssi...
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\va …
- 1,075
3
votes
Looking for references to study $U^p$ and $V^p$ spaces
Here is one fairly recent paper of Sebastian Herr (the second author in the paper you linked) and Timothy Candy in which they use $U^p$ and $V^p$ spaces. During a winter school in Obergurgl in 2022, H …
- 1,075
5
votes
0
answers
404
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&=0\qu …
- 1,075
1
vote
Accepted
Surjectivity of a class of integrals in dimensions two
An explicit counterexample
The answer to your question is: in general, no. Here I will show a counterexample.
As in the previous version of the answer, we set $\Omega=\mathbb R^2$, and for the sake of …
- 1,075
6
votes
1
answer
272
views
Analytic maps on Banach spaces: analyticity upgrade
Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such th …
- 1,075
3
votes
Analytic maps on Banach spaces: analyticity upgrade
I will summarize what has been said in the comments (thanks to Jochen Glueck for all his help).
The answer to the question is no, in general. What is going on is actually very simple.
Theorem. Let $E, …
- 1,075