Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,349
questions
0
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25
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An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
0
votes
1
answer
35
views
Rate of convergence of the minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
0
votes
1
answer
97
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
-1
votes
0
answers
27
views
Interchange of supremum and integral for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$
Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere
$$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$
Let ...
-3
votes
0
answers
298
views
Atiyah-Singer fake proof intuition?
Here is one formulation of the Atiyah theorem.
Fix a Riemannian $M$ and a pde $P: E \to F$ of bundles.
Let $\pi: S^*M \to M$ be the cotangent sphere.
Let $\sigma$ be the index of $P$ (so swap ...
0
votes
1
answer
69
views
Decay rate of minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
3
votes
0
answers
40
views
About the J-method of interpolation
In the classic text of J.Bergh and J.Lofstrom, Interpolation Spaces, the $J$-method of real interpolation defines a functor in the following way: For $0<\theta<1$ and $1\leq q\leq \infty$ we ...
2
votes
1
answer
63
views
Hausdorff-Lipschitz continuity of cone correspondence
Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let
\begin{equation}
f: \...
2
votes
0
answers
56
views
Schauder Frames in nuclear vector spaces
In recent years, the definition of frame has been extended to locally convex topological vector spaces (lcs) (1). In particular, let $X$ be a lcs and $X'$ its dual. A sequence $\big((x_n,y_n)\big)_{n\...
1
vote
0
answers
84
views
Hölder continuity for cocycles with respect to metrics
Let $T: X \to X$ be a uniform continuous (or Lipschitz continuous) on a compact metric space $(X, d_1)$. Assume that $Y$ is a Banach algebra and $f:X \to Y$ is a Hölder continuous with respect to the ...
5
votes
0
answers
220
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
-1
votes
0
answers
76
views
Harmonic Analysis Textbook Recommendation
I am looking for a textbook that covers much the same content as Stein's 'Singular Integrals' and 'Harmonic Analysis' textbooks and does so at similar pace and level, with similar organization of the ...
0
votes
0
answers
33
views
Projection measure and an integral formula for Lipschitz functions
Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
2
votes
1
answer
76
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Subspaces of $C_0$ on which $p$-norm are equivalent?
I have a question concerning the generalization of the following fact.
Let $E = C^0([0,1],\mathbb{R})$ endowed with the $\|.\|_\infty$ norm. One can show that if $F$ is a subspace of $E$ for which ...
0
votes
1
answer
76
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On weighted Fourier transforms
Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that
$$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$
...
3
votes
0
answers
89
views
Image of trace operator on $W^{2,1}(\mathbb{R}^2)$
It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$.
For ...
0
votes
0
answers
66
views
Let $(N_1,N_2)$ be commuting normal operators. Does an operator $V$ exist such that $N_i = V^{-1}T_iV$ for $(T_1,T_2)$ being contractions and $i=1,2$?
My PhD advisor and I are trying to prove something, and in our case, we would need to show, that two commuting normal operators $N_1, N_2 \in B(H)$ are similar to two commuting contractions $T_1, T_2 \...
0
votes
1
answer
104
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
0
votes
1
answer
76
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Continuous modification of tangent vector fields
Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
3
votes
1
answer
211
views
Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...
-1
votes
0
answers
38
views
Relating a modified convolution to the standard convolution for specific functions
I'm working with a convolution operation of the form:
$$
(u f) * v = \int_{\mathbb{R}} u(y) f(y) v(x - y) \, dy
$$
where $u$, $v$, and $f$ are admissible functions under the following conditions:
$u \...
-1
votes
0
answers
61
views
Prove that $C=\inf \left\{\|u\|^2\middle|u \in D^{1,2}\left(\mathbb{R}^N\right), \int_{\Omega}\left| u\right|^{2^*}=1\right\}$ is positive
We have Hardy inequality $$\int_{\mathbb{R}^N} \frac{u^2}{|x|^2} \, d x \leq C \int_{\mathbb{R}^N}|\nabla u|^2 \, d x \quad \forall u \in D^{1,2}\left(\mathbb{R}^N\right).$$
now we can use the Hardy ...
2
votes
0
answers
147
views
+100
Homeomorphically deforming one continuous function to another
Motivation:
A result of Klee Theorem 3.4 in [1], namely Klee's trick, implies that if $f,g:\mathbb{R}^d\to \mathbb{R}^n$ for any $d,n\in \mathbb{N}_+$ then there is a homeomorphism $\tilde{H}:\mathbb{...
6
votes
2
answers
417
views
Asymptotic behavior of the "Cauchy square" series
$\renewcommand{\ge}{\geqslant}\renewcommand{\le}{\leqslant}$
$\newcommand{\pa}[1]{\left( #1 \right)}$
Let us take $\alpha > 0$, $x_1 := \alpha$ and for any $n \ge \mathbb{N}$,
\begin{align*}
\boxed{...
2
votes
0
answers
114
views
Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
1
vote
0
answers
70
views
Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
3
votes
0
answers
193
views
Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
0
votes
0
answers
41
views
A follow up on "A bilinear estimate with a simple one-dimensional oscillatory integral kernel"
I am investigating the validity of the estimate
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,dz\leq C\,\epsilon\, \|f\|...
1
vote
1
answer
102
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
0
votes
0
answers
127
views
Notation $\le_{a,b,n,\ldots}$ in Analysis
In modern Analysis, especially Functional Analysis, one proves, or one uses inequalities of the form
$$F(X)\le_{a,\ldots,n}G(X).$$
The meaning of the subscripts in the inequality sign means that there ...
2
votes
0
answers
24
views
Monotonicity of the rank of finitely generated closed sublattices
Let $E$ be a Banach lattice. We consider finitely generated elements in the class of all closed sublattices of $E$. Let $F$ be a closed sublattice $E$. Then a subset $G \subseteq F$ is called ...
2
votes
0
answers
69
views
What is known about $\operatorname{gap}(A) = \|A\| - r(A)$ for bounded operators on Hilbert spaces?
The gap of a bounded linear operator on a Hilbert space is defined as $$\operatorname{gap}(A) := \|A\| - r(A),$$
where $r(A)$ denotes the spectral radius of $A$. A natural question to ask is - for ...
3
votes
0
answers
111
views
Explicit basis of symmetric harmonic polynomials
An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...
0
votes
0
answers
53
views
Criteria giving sufficient conditions for a Borel measure to have compact support
I am interested in criteria that guarantee that a Borel probability measure has compact support.
I outline two below and I am hoping to gather more as answers (if they exist).
The first sufficient ...
0
votes
0
answers
42
views
Johnson-Lindenstrauss type result for matrix factorization
The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
2
votes
0
answers
54
views
Basis vectors using anti-commuting operators?
Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
0
votes
0
answers
35
views
When does the Hermite series converge pointwise and when is it uniformly bounded?
Let $\gamma$ denote the standard Gaussian measure on the real line, and consider $f \in L^2(\gamma)$. Since the Hermite polynomials $\{H_n\}_{n \geq 0}$ are a complete orthonormal system, we may ...
1
vote
1
answer
86
views
Bounding supremum norm in terms of gradient L2-norm using a Poincare-like inequality
Suppose $f$ is a Lipschitz continuous real-valued function over a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary, and let $\overline{f} := \frac{1}{|\Omega|}\int_\Omega f(x) dx$. Is ...
6
votes
1
answer
198
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 ...
0
votes
0
answers
57
views
Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
3
votes
0
answers
134
views
Interchange limit and supremum of functionals over a bounded convex set
Let $(H, \langle\cdot,\cdot\rangle)$ be a separable real Hilbert space and $B\subset H$ be (nonempty) convex and bounded, and suppose that $(\alpha_k)\subset H$ is a sequence for which the limit $\...
2
votes
1
answer
269
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
0
votes
0
answers
79
views
Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
0
votes
0
answers
69
views
Schauder basis of L^p on general domains
Let $\Omega$ be a non-empty regular compact subset of $\mathbb{R}^n$; i.e. the closure of its interior is itself; for some $n\in \mathbb{Z}^+$. Let $1\le p<\infty$. When does the space $L^p(\...
0
votes
0
answers
41
views
Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?
We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-...
1
vote
1
answer
42
views
Lower bound the best $\alpha$-Hölder constant of a convolution
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
1
vote
2
answers
80
views
Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
6
votes
0
answers
241
views
Applications of Banach space homology
There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
0
votes
0
answers
48
views
A problem about how to understand the existence of derivative of level set in Mountain-pass theorem
I'm confused about the Mountain pass theorem in Lemma4 of here.
Background :
$$
\begin{gathered}
I_\lambda(u)=\frac{1}{2} \int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g-\frac{\lambda}{2 m} \log ...
0
votes
0
answers
80
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 $\alpha$ ...