All Questions
12,937 questions
0
votes
0
answers
45
views
Fractal dimension using wavelets [closed]
I'm trying to estimate the fractal dimension of a function.
I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation
$$
\alpha = -...
1
vote
0
answers
37
views
Isomorphism between weighted Sobolev spaces via Laplace operator
My question is on weighted Sobolev spaces $W^{k,2}_{\delta}(\mathbb R^n)$ and whether I can find a good reference that states when is the following mapping
$$ \Delta: H^2_{\delta}(\mathbb R^n) \to L^...
- 4,153
3
votes
3
answers
399
views
Huygens' principle or finite speed of propagation?
I am reading the paper Large global solutions for energy supercritical nonlinear wave equations on $\mathbb{R}^{3+1}$ by Krieger and Schlag and am confused by one of their steps.
For context, $v(t,r)$ ...
- 870
-1
votes
0
answers
45
views
Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
3
votes
1
answer
76
views
Tangential Sobolev spaces
Let $Ω⊂R^n$ be a smooth domain, define $U_s=\{x∈Ω | d(x,∂Ω)<s\}$; let $f∈W^{1,p}(Ω)∩W_{\mathrm{loc}} ^{2,p}(Ω)$; let $v$ be the unit normal to $Ω$; consider $v$ to be smooth with bounded ...
0
votes
0
answers
37
views
Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
- 455
4
votes
1
answer
282
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
2
votes
0
answers
116
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
-3
votes
0
answers
75
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
9
votes
1
answer
301
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
0
votes
0
answers
67
views
$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable
Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported.
I am wondering if we can ...
- 91
1
vote
0
answers
35
views
Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem:
$$
\begin{cases}
- \Delta u + u = u |u|^{p - 2}
&\text{in} ~ \mathbb{R}^N;
\\
u (x) \to 0 &\text{as} ~ |x| \to \infty,
\end{cases}
$$
where $N \geq 2$...
- 144
12
votes
1
answer
396
views
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 24.2k
0
votes
0
answers
89
views
+100
Uniqueness of bubbling points in Struwe's global compactness theorem
I am reading the following paper of Struwe in which he proves the following result:
Proposition 2.1:
Let $n\geq 3$, $\lambda \in \mathbb{R}$ and $\Omega$ be a smoothly bounded domain in $\mathbb{R}^{n}...
- 537
3
votes
1
answer
388
views
Do we have Pohozaev's identity on compact manifolds without boundary？
Recently I got to know about Pohozaev's identity, and I calculated several examples. The basic idea is multiplying $x \cdot \nabla u$ on both sides of the equation, but I noticed that all the ...
- 809
3
votes
0
answers
88
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 73
4
votes
1
answer
64
views
Mapping properties of the Schrödinger semigroup
The Schrödinger semigroup $e^{t(-\Delta +V(x))}$ for Kato class potentials is fairly well-understood. A classical reference is the AMS paper "Schrödinger Semigroups" by Barry Simon. I was ...
- 1,045
2
votes
0
answers
229
views
A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989
3
votes
0
answers
91
views
About BMO space on smooth open bounded domain
Let $\Omega$ be any open domain in $\Bbb R^d$.
Define the $\text{BMO}(\Omega)$ space as
$$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\},
$$
...
- 1,101
2
votes
0
answers
17
views
On compact embeddings in weighted Riesz potential spaces
I wonder if there is any references for the study of the following type of spaces
$$ X_{\delta,\alpha}=\{ u\in L^2_\delta(\mathbb{R}^n):\, u= (-\Delta)^\alpha f \quad\text{for some}\quad f\in L^2_{\...
- 4,153
3
votes
0
answers
67
views
Effective action of unbounded operators on subspaces outside their domains of definition
Consider a densely defined, self-adjoint operator
$$
H: \mathcal{D} \rightarrow \mathscr{H}.
$$
Assume for simplicity that $H$ is nonnegative.
We want to effectively restrict this operator $H$ to a ...
- 133
2
votes
0
answers
80
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
3
votes
0
answers
78
views
Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$
In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
0
votes
1
answer
159
views
Weak convergence of $f(x,e^{itx})$
This is the desired result (what I want to prove):
$$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$
Given that $f\in C([a,b]\times\{e^{i\...
- 231
11
votes
0
answers
325
views
+50
Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution.
In 2015, when the second edition of the Scottish Book with updates and commentary on ...
- 13k
1
vote
1
answer
153
views
Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 113
1
vote
0
answers
88
views
Density of a subset of Schwartz space in the fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
4
votes
0
answers
91
views
Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 825
0
votes
0
answers
46
views
Amenability of locally convex algebras
Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.).
Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
- 2,605
2
votes
0
answers
68
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
53
views
Can one explicitly define a right inverse for a convolution operator on the space of entire functions?
A result of Meise and Taylor in 1988 shows that every non-zero convolution operator on the Frechet space $H(\mathbb{C})$ of all entire functions on $\mathbb{C}$ has a continuous linear right inverse $...
0
votes
1
answer
239
views
Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]
Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
1
vote
0
answers
58
views
duality of sobolev spaces. Representation of elements in the dual
I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
2
votes
1
answer
239
views
Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 825
0
votes
0
answers
18
views
Third order estimate for linear elliptic equations
Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
- 455
0
votes
0
answers
87
views
Curl-Div equation with singular matrix
I want to solve the equation:
$$
\begin{cases}
\nabla \times (A \mathbf v)=f, \quad x\in \Omega \\
\operatorname{div}(\mathbf v)=0,
\end{cases}
$$
where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
16
votes
1
answer
977
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
0
votes
0
answers
62
views
Characterization of duals of Sobolev space
Proposition 8.14. in Brezis states that:$(W_0^{1,p} (Ω))^*=W^{-1,p^*} (Ω)$ and we have the representation:
$∀ F∈(W_0^{1,p} (Ω))^* ∃ f_0...f_n ∈L^{p^*} (Ω)$ such that $∀ u∈W_0^{1,p}(Ω)$
$F(u)=∫_Ω ...
0
votes
1
answer
126
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,153
4
votes
1
answer
261
views
Are renormalizability and the criticality of a PDE synonymous?
In the physics literature a quantum field theory is qualitatively classified as renormalizable, super-renormalizable, or non-renormalizable. This heuristic is based on how many Feynman diagrams ...
- 721
5
votes
1
answer
208
views
Compactness in trace class operators space
Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$.
Are there easy ...
- 59
0
votes
0
answers
90
views
How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?
I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak*
-Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
- 113
1
vote
0
answers
148
views
integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 531
4
votes
0
answers
101
views
There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions
Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$.
Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
- 3,669
3
votes
1
answer
128
views
Comparing two different principles of premeasure-to-measure extension
It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for ...
- 133
0
votes
1
answer
171
views
Is the evolution family self-adjoint?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
\newcommand{\qtext}[1]{\quad\text{#1}}
\newcommand{\qtextq}[1]{\quad\text{#1}\quad}
$
I am reading Roland Schnaubelt's survey ...
- 825
3
votes
0
answers
158
views
Gowers' dichotomy for quotients
Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable.
A ...
- 4,461
2
votes
0
answers
194
views
Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
- 20.2k
0
votes
0
answers
77
views
Nice formula for powers of modified Bessel function
Let $K_\nu(z)$ be the modified Bessel function of second kind. I am looking the geometric series
$$1+aK_v+(aK_v)^2+(aK_v)^3...$$
I know there are formula for product of two such functions. I would ...
- 275
3
votes
1
answer
189
views
Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces
An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such ...
- 33