All Questions
10,199 questions
1
vote
0
answers
153
views
A transformation game for natural numbers?
Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS ...
- 3,074
1
vote
0
answers
130
views
Brezis-Kato theorem
Let $n\geq 3$, and $u$ satisfies
$$
-\Delta u=K(x)u^{\frac{n+2}{n-2}}
\quad x\in B_1\setminus \{0\},
$$ where
$|K(x)|\leq A$ in $B_1\setminus \{0\}$, and
$u\geq 0$ in $B_1\setminus \{0\}$.
Can we ...
- 471
1
vote
0
answers
76
views
Positive definite function on the upper half line and its integral expression
By Hausdorff-Bernstein-Widder theorem, any completely monotonic function on the half line $\mathbb{R}_{\geq 0}:=[0,\infty)$
is given by the Laplace transform of a positive measure on $\mathbb{R}_{\geq ...
- 63
1
vote
1
answer
90
views
Positive definite but not completely monotonic function on the upper half line
By Hausdorff-Bernstein-Widder theorem, any completely monotonic function on the half line $\mathbb{R}_{\geq 0}:=[0,\infty)$ is given by the Laplace transform of a positive measure on $\mathbb{R}_{\geq ...
- 63
0
votes
1
answer
185
views
Spectrum of a product of a symmetric positive definite matrix and a positive definite operator
Let $\mathbf H$ be an infinite dimensional Hilbert space.
I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
- 63
1
vote
0
answers
120
views
Formula for the kernel of an operator
Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...
- 521
0
votes
0
answers
138
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 791
3
votes
0
answers
124
views
Leibniz rule bound for the inverse of the Laplacian?
Let $f, g \in L^2[\mathbb{T}^2]$ be real-valued functions without zero modes. That is, $\int_{\mathbb{T}^2}f=\int_{\mathbb{T}^2}g=0$. Here, ${\mathbb{T}^2}$ is the $2$-dimensional torus $[\mathbb{R}/\...
- 3,477
4
votes
0
answers
126
views
Darboux integral for non-polynomial ODEs
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$
\dot{x}_j=f_j(x)=\sum_{i_1,\dots,i_n=1}^d a_{i_1,\dots,i_n}^j x_1^{i_1}\dots x_n^{i_n} \quad \forall j=1,\ldots,n
$$
we define ...
- 247
1
vote
1
answer
355
views
Hilbert–Pólya conjecture with Grommer inequalities?
The Grommer inequalities are equivalent to RH and formulated on page 20 of Conrey - Riemann's hypothesis:
Let
$$\Xi(t) := \xi(1/2+it).$$
Then RH is equivalent to : All zeros of $\Xi(t)$ are real. The ...
- 3,074
0
votes
1
answer
153
views
An integral Minkowski inequality for the quasi-Banach case?
The so called integral Minkowski inequality claims that for suitable positive functions $f$ defined on the product of two measure spaces $ (X\times Y, \mu \times \nu) $ and $p\geq 1$ we have that
$$ \...
0
votes
0
answers
355
views
On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 790
1
vote
0
answers
82
views
Commutator of self-adjoint operators and $C^1$-type formula
Let $\mathcal{H}$ be a (complex) Hilbert space.
Let $H$ be a self-adjoint operator on $\mathcal{H}$ with dense domain $\mathcal{D}(H) \subset \mathcal{H}$, generating the unitary one-parameter ...
- 71
2
votes
1
answer
645
views
Reference request: inverse of differential operators
I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).
As an example ...
- 721
4
votes
1
answer
121
views
Ext groups of locally convex topological vector spaces
Suppose I work over $\mathbb{C}$. Is it known for which locally convex topological vector spaces $V$, we have $\text{Ext}^i(V, \mathbb{C})=\{0\}$ for all $i>0$, working with the type of Ext groups ...
- 1,180
0
votes
1
answer
145
views
Renorming on a separable Banach space
Let us consider the sequence space $c_0$ with the equivalent norm
$$\Vert x \Vert^2 = \max_{i\ge1} \vert x^i \vert^2 + \sum_{i=2}^{\infty} 2^{-i+1} \vert x^i \vert^2 $$
for $x=(x^1,x^2,\ldots)\in c_0$....
- 85
1
vote
1
answer
197
views
Does convolution with heat kernel converge to pointwise evaluation?
Let $G(t, x) := \frac{1}{\sqrt{4 \pi t}} \exp\left( -\frac{x^2}{4 t }\right)$ for all $(t, x) \in (0, T) \times \mathbb{R}$ be the fundamental solution to the heat equation $\partial_tu = \partial_{...
5
votes
1
answer
311
views
Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator
$$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem ...
- 852
2
votes
1
answer
139
views
Can a chaotic trajectory solve an algebraic equation?
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$\dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n$$
we ...
- 247
2
votes
1
answer
61
views
$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
1
vote
0
answers
37
views
Unique smallest degree algebraic solution to polynomial ODE
Let's assume we are given a degree $d$ polynomial VF as a map $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$
$$f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a^j_{i_{1},\dots,i_{n}}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$$...
- 247
3
votes
0
answers
140
views
Does the Kato-Ponce estimate hold on manifolds?
Recall the Kato-Ponce estimate for fractional powers of the operator $J = (1-\Delta)$,
$$ \| J^s(fg) \|_{L^r} \lesssim \| J^s f \|_{L^{p_1}} \| g \|_{L^{q_1}} + \| J^s g \|_{L^{p_2}} \| f \|_{L^{q_2}},...
- 914
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
2
votes
2
answers
200
views
If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised
Let us consider the Fréchet space $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ of real-valued, periodic smooth functions.
That is, $f_n \to f$ in $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ if $f^{(m)}_n$ ...
- 3,477
3
votes
1
answer
108
views
$L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality
For each $n \in \mathbb{N}$, the Hermite function $\psi_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by
\begin{equation}
\psi_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}...
- 3,477
2
votes
0
answers
144
views
Examples of topologically non trivial complete submanifolds in infinite dimensional Banach spaces
In infinite dimensional Banach spaces, many analogies of classical sets are topologically trivial ( even contractible). E.g., infinite dimensional spheres are contractible by Y. Benyamini, Y. ...
- 593
1
vote
0
answers
35
views
Finding bounds for a "smooth" mapping $u : [0,T] \to \mathcal{S}(\mathbb{R})$ in terms of Schwartz seminorms
Let $u(t,x) : [0,T] \times \mathbb{R} \to \mathbb{R}$ be a smooth function with the following properties:
Partial derivatives of $u(t,x)$ with respect to $t$ up to any order are Schwartz functions of ...
- 3,477
2
votes
0
answers
83
views
Singular integral operators acting on Zygmund class
It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies
$$\sup_{0<R<\...
- 21
1
vote
0
answers
109
views
$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$u' - \Delta u = 0 \quad\text{on $\Omega$}$$
$$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$
$$u(t=0)=u_0\quad\text{on $\Omega$}$$
where $\Omega$ is a bounded ...
- 93
4
votes
1
answer
259
views
The real and the imaginary part of a vector
In an infinite-dimensional Banah space $(X, \|\cdot\|)$ with a countable Schauder basis $\{x_n\}$, define:
$$
F_r: \operatorname{Span}(\{x_n\}) \rightarrow \operatorname{Span}(\{x_n\}), \hspace{0.3cm} ...
- 307
3
votes
1
answer
428
views
Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I ...
- 3,477
0
votes
1
answer
141
views
Infimum of norms of elements in a hyperplane
In a Banach space X, given a norm one bounded linear functional $f$ and $c\in \mathbb{C}\backslash \{0\}$, define $H = \big\{ x\in X \,\vert\, f(x) = c\big\}$ and $\inf H$ = $\inf_{h\in H} \|h\|$.
Is ...
- 307
2
votes
0
answers
103
views
Schwartz kernel theorem for restricted operators
Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...
- 1,171
3
votes
1
answer
219
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 657
0
votes
1
answer
241
views
Norm functions induced by convex bodies
Given a centrally symmetric convex body $K$ in the plane (with smooth boundary), it is easy to see that there exists a norm function $g:\mathbb{R}^2\to \mathbb{R}_{\geq 0}$ for which $K$ is the unit ...
- 191
1
vote
1
answer
262
views
Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support
In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the
Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V_n)_n$ ...
-2
votes
1
answer
216
views
Inverse of Sobolev interpolation inequality : $\lVert u \rVert_2 \lVert \Delta u \rVert_2 \leq C\lVert \nabla u \rVert_2^2$?
If $u : \mathbb{T}^3 \to \mathbb{R}$ is a smooth function on the $3$-dimensional torus $\mathbb{T}^3$, I wonder it is possible to reverse the Sobolev interpolation inequality in the sense that
\begin{...
- 3,477
4
votes
2
answers
904
views
Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?
Trying to find and answer to this question, I have encountered two more-studied problems.
The first is to find when a Banach space admits an equivalent uniformly convex norm. The answer is that for ...
- 1,955
3
votes
2
answers
147
views
Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
- 300
1
vote
0
answers
95
views
Are the sum and product of nonlinear compact operators compact?
In the 'interactive' proof of Lomonosov's Theorem about hyperinvariant subspaces (in the book Hilbert Space Operators, A Problem Solving Approach), one is asked to prove the compactness of the ...
- 11
1
vote
1
answer
180
views
Conditioning a $\mathrm{C}^*$-algebra state with infinite precision
This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success.
Let $\mathcal{A}$ be a unital $\...
- 1,037
4
votes
0
answers
119
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
0
votes
0
answers
417
views
Spectral theorem for commuting operators
Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation ...
- 3,535
1
vote
0
answers
164
views
Reference on spectral theory of self-adjoint operators
I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing an operator $L$ over smooth functions (and extended to an self ...
- 13
7
votes
0
answers
162
views
Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
- 1,805
0
votes
0
answers
134
views
Are there any books or literature on norms over measure space?
Consider the space of signed measures over some abstract space, we know the total variation norm makes the space Banach (I guess). So are some other norms. Are there some books or literature studying ...
0
votes
2
answers
197
views
Convergence of the infima of convex functions on $\mathbb{R}^m$
Any thoughts on proving the following statement, which is a generalization of the result in convergence of the infima of convex functions from domain $\mathbb{R}$ to $\mathbb{R}^m$ and also Theorem 1 ...
6
votes
3
answers
551
views
Hahn-Banach Theorem for convex polytopes and their supporting hyperplanes
A polytope in $\mathbb R^n$ is the convex hull of a nonempty finite set in $\mathbb R^n$.
Let $C$ be a polytope in $\mathbb R^n$.
We shall say that a hyperplane $H\subseteq \mathbb R^n$
$\bullet$ ...
- 42k
1
vote
0
answers
32
views
Rätz orthogonality and involution
In the Rätz’s sens of orhtoganality, can we find an exemple of an involution u(different to -Id)such that x orthogonal to y then x orthogonal to u(y)
0
votes
1
answer
228
views
Norm equivalence in finite dimensions - is the equivalence "universal" if the dimension is fixed?
I am aware that in a finite dimensional vector space, any two norms are equivalent.
However, I cannot really figure out how "universal" the equivalence constants are.
To be specific, let us ...
- 3,477