All Questions
9,780 questions
2
votes
0
answers
81
views
Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 434
0
votes
1
answer
117
views
How to understand the unique continuation result
Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm
$$
\|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}.
$$
Suppose $K(x) \in C^1\left(\mathbf{R}^...
- 471
3
votes
0
answers
105
views
Maximal-type inequality 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 ...
- 890
1
vote
1
answer
112
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
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852
0
votes
1
answer
90
views
For any $p, q \in [1,\infty]$ and $s \in (0,\infty)$, can we find some $f \in L^q - W^{s,p}$?
Sobolev inequalities show us when we can embed a Sobolev space into another.
However, I wonder if these inclusions are always proper.
More specifically, let $\Omega \subset \mathbb{R}^n$ be a bounded ...
- 3,477
2
votes
1
answer
214
views
Are projective tensor products left-exact if one considers only maps of norm at most 1?
Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...
- 843
2
votes
0
answers
132
views
Convergence of integral operators' inverses
Let $K$ be a positive definite integral operator with continuous kernel $K(x,y)$ defined by
$$
Kf(x) = \int_0^1 K(x,y) f(y) \, dy.
$$
Let $K_n$ denote the matrix $K(x_i, x_j)$ with $x_i = i /n$ and ...
- 620
6
votes
1
answer
796
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
- 128k
2
votes
1
answer
79
views
Hausdorff-Lipschitz continuity of cone correspondence
Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let
\begin{equation}
f: \...
- 55
1
vote
1
answer
370
views
Bounding supremum norm in terms of gradient L2-norm using a Poincare-like inequality
Suppose $f$ is a Lipschitz continuous real-valued function over a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary, and let $\overline{f} := \frac{1}{|\Omega|}\int_\Omega f(x) dx$. Is ...
- 148
3
votes
2
answers
183
views
Dimension of spectral projection subspaces under strong convergence of operators
I have a possibly simple question regarding estimating bounds on spectral projection subspace.
Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the ...
- 633
0
votes
1
answer
102
views
On weighted Fourier transforms
Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that
$$ \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.$$
...
- 4,115
3
votes
2
answers
280
views
Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...
- 421
4
votes
0
answers
108
views
Larger possible chain of closed subspaces in the dual of a Banach space
In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces.
My question is the following. If $X$ is an ...
- 153
2
votes
1
answer
154
views
Are these two norms on localized versions of $L^p_q$ equivalent?
$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued ...
- 825
2
votes
1
answer
112
views
On compactly supported functions with prescribed sparse coordinates
Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
- 4,115
1
vote
1
answer
152
views
Points in the Stone Cech compactification are intersection of open sets
Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
0
votes
0
answers
235
views
Analogue of $\ell^2(X)$ over an arbitrary Banach ring
Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...
- 1,485
0
votes
1
answer
143
views
Is the space $C_0^{k}(\Omega)$ a Montel space?
I asked this question in the MathStackExchange, but I think I'm not get any answer.
I'm trying to find a reference for the following result:
Theorem: Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ ...
- 509
2
votes
0
answers
75
views
Regularity of solutions to an elliptic boundary value problem
Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
- 969
60
votes
23
answers
108k
views
A good book of functional analysis [closed]
I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics)....
6
votes
1
answer
285
views
Distinguishing the Besov and Triebel-Lizorkin spaces
Theorem 2.3.9. in Triebel's Theory of Function Spaces states that the Besov space $B^{s_1, p_1}_{q_1} (\mathbb R^d)$ coincides with the Triebel-Lizorkin space $F^{s_2, p_2}_{q_2} (\mathbb R^d)$ if and ...
- 301
3
votes
2
answers
616
views
A problem about how dominated convergence is used in the analysis of variation
I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...
- 809
5
votes
2
answers
458
views
Logarithm of a bounded operator
Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that
$
A=\exp L
$...
- 16.2k
3
votes
1
answer
79
views
Can a lift satisfy Chen's relation, geometric condition but not be a rough path?
Let $(X,\mathbb X):[0,1]^2\to \mathbb R^d\oplus\mathbb R^{d\times d}$ satisfy the following four properties:
\begin{align}
&X_{s,t}=X_{0,t}-X_{0,s}\\
&\sup_{t\neq s}\frac{|X_{s,t}|}{|t-s|^\...
- 1,914
2
votes
1
answer
186
views
Koopman operators on $L^p(X)$
On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
3
votes
0
answers
103
views
How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
- 1,955
1
vote
2
answers
90
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 (...
- 825
0
votes
1
answer
121
views
A simple bilinear estimate
Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that
$\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$.
Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$.
What is the optimal value of $t=t(\...
- 852
3
votes
1
answer
163
views
Rescaling Fourier coefficients of a continuous function by a bounded sequence
This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
- 404
1
vote
0
answers
64
views
embedding spaces of probability measures to function spaces
Let $X, Y$ be Banach spaces. I'm considering a bounded linear functional $g:X\to Y$ and its lift $g_\sharp: \mathcal{P}(X)\to \mathcal{P}(Y)$.
I want to consider the inverse of $g_\sharp$ in some ...
0
votes
0
answers
89
views
Maximal function on mixed $L^{p}$
Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is
$$
\Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
- 31
0
votes
1
answer
77
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 ...
- 434
0
votes
1
answer
80
views
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 ...
- 434
1
vote
1
answer
101
views
Image of a complete topological group under open and surjective map is complete?
A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff.
Question.
Let $G$ be a complete topological group and let $H$ be a topological ...
- 532
0
votes
0
answers
99
views
Dual of closure
Currently I'm studying about abstract interpolation theory for my research. One of the basic ways to construct new interpolation spaces, given an interpolation space $E$ with respect to a compatible ...
6
votes
1
answer
249
views
Syndetic sets and Banach limits: reference request
First of all, let us give a few definitions. Suppose that $A$ is a subset of natural numbers. We say that $A$ is syndetic if there is a constant $M$ such that every set of $M$ consecutive natural ...
- 7,193
5
votes
1
answer
221
views
In what sense does the Laplacian on compact intervals converge to one on all of $\mathbb{R}$?
I guess this topic may have been addressed somewhere but I cannot really find a reference myself, so I ask here.
For each $N \in \mathbb{N}$, consider the Laplacian $\Delta$ on the interval $[-N,N]$ ...
- 3,477
3
votes
0
answers
282
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 $\...
- 463
1
vote
1
answer
133
views
A question about the maximal function
Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
- 155
2
votes
1
answer
128
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 1,245
3
votes
0
answers
207
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 ...
- 31
0
votes
1
answer
192
views
A continuous injection from the Hilbert cube to the real line?
Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
- 3,104
2
votes
0
answers
138
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 $L^2([a,b]\times S^2)$ ...
- 969
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
2
votes
1
answer
173
views
Difference in essential spectrum between Schrodinger operators
I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
- 633
0
votes
1
answer
118
views
Minimal norm problem whose unknown is an operator
Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that
$$
f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2
$...
- 115
3
votes
0
answers
128
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 ...
- 989
2
votes
2
answers
308
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
- 113
1
vote
1
answer
171
views
How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?
In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...
- 11