All Questions
12,935 questions
0
votes
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74
views
Operator globally hypoelliptic
An operateor $T$ is globally hypoelliptic if :
$u\in S'(\Bbb R^n),Tu\in S(\Bbb R^n)$ imply $u\in S(\Bbb R^n)$.
My question why if $u\in L^2(\Bbb R^n): Tu =\lambda u$. Then $u\in S(\Bbb R^n)$.
where $\...
- 531
2
votes
1
answer
160
views
Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term
I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying
$$
\begin{...
- 35
0
votes
0
answers
66
views
convolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?
Let $U$ and $E$ ...
1
vote
1
answer
111
views
How to show such result for generalized $ O(|x|^{-1/2}) $ function?
Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
- 1,219
5
votes
1
answer
108
views
Uniqueness of constructed solutions to the Helmholtz equation
My question is regarding the inhomogeneous Helmholtz equation on $\mathbb{R}^3$ with real wavenumber $k$ and outgoing radiation condition
\begin{equation}
\Delta u + k^2 u = - f \quad \text{and} \quad ...
1
vote
0
answers
56
views
Extensions of functions with bounded $\int \lvert\operatorname{div} A(\nabla u)\rvert\, dm$
Consider the annulus $\mathcal A:= B(0,2)\setminus B(0,1)$ in $\mathbb R^n$, $n\geq2$ and the divergence form elliptic operator $\operatorname{div}(A\nabla \cdot)$ on $\mathcal A$ where $A$ is a ...
- 393
0
votes
2
answers
167
views
Is a signed measure $\mu$ on $\mathbb{R}^d$ characterized by the transform $\mathcal{L}_\mu (\lambda ):=\int e^{\langle \lambda,x\rangle }\mu (dx)$?
In the book "Probability Theory" by Achim Klenke there's the following theorem: a finite measure $\mu$ on $[0,\infty )$ is characterized by its Laplace transform $\mathcal{L}_\mu(\lambda):=\...
- 15
0
votes
0
answers
143
views
A Poincaré inequality holds for $p>2$ but fails for $p\leqslant 2$
I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).
Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset ...
- 69
1
vote
0
answers
64
views
Extension of meromorphic distribution
Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
- 3,799
1
vote
1
answer
126
views
Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as
$$
J_f(x)=S(x)\Lambda(x) {S(x)}^{-1},
$$
where the ...
- 19
9
votes
1
answer
639
views
Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
- 807
0
votes
3
answers
278
views
A generalisation of Tchebychev inequality
Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$.
Is it true that exists $ u$ any real function, and $a,b$ monotone ...
- 4,074
0
votes
1
answer
80
views
Orthogonal space of polynomials
Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and
$$
\int_0^{+\infty} f(t) t^n ...
1
vote
1
answer
80
views
Does local gradient collinearity imply factorization
Assume that we have two analytic functions $f(x,y)$ and $g(x,y)$ defined near $(0,0)$ on the plane. Let us also assume that $\nabla f =k \nabla g$, where $k=k(x,y)$ is also analytic and $\nabla f(0,0)\...
- 728
1
vote
0
answers
105
views
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
- 29
19
votes
3
answers
1k
views
What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?
A colleague asked me the following question:
"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"
This ...
- 366
2
votes
1
answer
126
views
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 ...
- 125
-1
votes
1
answer
286
views
Check an equation on the Heisenberg group $H_1$
The Heisenberg group $H_1$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right); \quad \forall z,w \in \mathbb C\,...
- 650
0
votes
1
answer
97
views
Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms
The paper by Eckmann-Epstein proves that Schwinger functions at "coinciding points" uniquely defines "time-ordered products".
In physics, these "time-ordered products" ...
- 3,477
7
votes
2
answers
592
views
Prove that the following function is positive
Consider the following function:
$$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right)
$$
This is Mehler's kernel, and can be ...
- 90
1
vote
1
answer
110
views
Looking for definition of function spaces appearing in article of DiPerna & Lions
I am looking for the definition of various function spaces appearing in the following article, preferably with references to other sources where such spaces are discussed in greater detail:
Article: ...
- 21
2
votes
0
answers
107
views
Finite dimensional manifolds as subspace of $\mathbb{R}^\mathbb{N}$
For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the ...
- 275
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,115
16
votes
1
answer
786
views
The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
- 52.3k
69
votes
3
answers
12k
views
Nonconvexity and discretization
Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem.
Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
- 21.3k
0
votes
0
answers
45
views
Mean value property for fractional laplacian
I just started reading about fractional Laplacian. I am curious on the following questions
Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value ...
- 41
2
votes
0
answers
56
views
Convergence of conformal metrics with prescribed curvature
We know that for any function $K: \mathbb{D} \to \left[-a, -b\right]$, where $a, b > 0$, there is a unique metric $h$ on the disk $\mathbb{D}$ which is conformal to $dz^{2}$, and has curvature ...
- 63
4
votes
0
answers
262
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 ...
- 1,755
2
votes
1
answer
321
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
- 4,074
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 41
2
votes
0
answers
71
views
Complemented subspaces of $\mathcal s$
Crossposted from Math Stack Exchange
It is well known that a nuclear Fréchet space $X$ is isomorphic to a complemented subspace of $\mathcal s$ (the space of rapidly decreasing sequences) if and only ...
- 655
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
6
votes
2
answers
290
views
If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?
In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space.
I wonder if the same result holds valid in infinite dimensions. More ...
- 3,477
1
vote
0
answers
40
views
Uniform bound for coefficients of fundamental forms of unbounded surface
Recently I am using the Gauss-Weingarten relations to transform an equation defined on a domain $\Omega\in\mathbb{R}^3$ into a form expressed in local coordinates when it approaches the boundary $\...
- 71
11
votes
1
answer
309
views
Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?
In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
- 60.6k
3
votes
1
answer
161
views
Definitions of weak solutions for quasilinear wave equations
I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
- 89
3
votes
0
answers
79
views
Continuity of disintegrations in non locally compact spaces
Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
- 63
0
votes
0
answers
55
views
Status of generalization of timelike tube theorem to algebras of causal completions
The timelike tube theorem states that the additive algebra $A_{\text{add}}(U)$ of operators in a spacetime region $U$ is equal to the additive algebra $A_{\text{add}}(E(U))$ of the timelike envelope $...
user528837
10
votes
2
answers
548
views
Wick rotation for Laplace and wave equations
I have seen Wick rotation used to describe the relationship between the heat and Schrodinger equations. That is, if $u(t,x)$ solves the heat equation then $v(t,x):=u(it,x)$ solves the Schrodinger ...
- 1,914
6
votes
2
answers
644
views
Explicit form of this unitary transformation
Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if ...
- 3,535
2
votes
0
answers
137
views
Holder-Besov space and time continuity
Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...
- 573
2
votes
1
answer
272
views
Decompose a function into a bounded part and a Lipschitz part
Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that
$$
\sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty.
$$
Are there functions $g,h: \mathbb R^d \...
- 825
23
votes
5
answers
8k
views
Why do we have two theorems when one implies the other?
Why do we have two theorems one for the density of $C^{\infty}_c(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$ and one for the density of $C^{\infty}_c(\Omega)$ in $L^p(\Omega)$? with $\Omega$ an open subset ...
1
vote
0
answers
63
views
$\operatorname{ker}(q_I \otimes^{\text{min}} q_J) $ is a primal ideal of $\mathcal{A} \otimes^{\text{min}} \mathcal{B}$
In the proof of Theorem $4.1$ of the paper titled continuous bundles of $C^{\ast}$-algebras and tensor products following result is mention with a reference to Proposition $3.3$ of the paper "A. ...
- 1,115
7
votes
0
answers
295
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 ...
- 175
4
votes
1
answer
97
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_{...
- 938
2
votes
2
answers
242
views
A Inequality in the paper by Kenig, Ponce and Vega
I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle",
...
- 87
0
votes
0
answers
89
views
Weakly compact set
I want to show that if the set
$$
\big\{u \in L^{q}([0, n] ; X): u(t) \in \phi(t, x(t)), t \in[0, n]\big\}
$$
is weakly compact, then the set
$$
\mathcal{S}_{\phi}(x)=\Big\{u\in L_{loc}^{q}(\mathbb{R}...
- 39
0
votes
1
answer
114
views
Ball in separable Banach space has positive Gaussian measure
I have (presumably non-degenerate) Gaussian $\mu$ over separable Banach space $X$. I would like to prove that for any ball of radius $r$ centered at $x$, $\mu(B_r(x))$. I know how to prove this in ...
- 379
2
votes
1
answer
232
views
Banach spaces locally having a basis
The $\mathcal{L}_p$-spaces ($1\leq p \leq \infty$) are Banach spaces $X$ such that there exists a constant $\lambda$ so that every finite dimensional subspace $E$ of $X$ is contained in another ...
- 4,461