All Questions
3,599 questions with no upvoted or accepted answers
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
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
2
votes
0
answers
152
views
Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?
The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
- 989
2
votes
0
answers
70
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\...
- 655
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
2
votes
0
answers
28
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 ...
- 467
2
votes
0
answers
78
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 ...
- 165
2
votes
0
answers
60
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:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
- 1,305
2
votes
0
answers
88
views
An example of an $\mathcal{L}_\infty$ Banach space with property p-(V) and without property (V)
Here are the definitions for property $p$-$(V)$ and property $(V)$.
A Banach space $X$ has property $(V)$ if and only if every unconditionally converging operator $T$ from $X$ to any Banach space $Y$ ...
2
votes
0
answers
103
views
What $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?
I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
- 1,231
2
votes
0
answers
35
views
Continuity of Kernel Mean Embeddings
Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
- 161
2
votes
0
answers
138
views
Sufficient initial conditions for "non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 71
2
votes
0
answers
147
views
About normal states in abstract von Neumann algebras
In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16
but this was state only for concrete von Neumann algebras (because ...
- 424
2
votes
0
answers
56
views
Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...
- 93
2
votes
0
answers
143
views
Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
2
votes
0
answers
111
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
- 825
2
votes
0
answers
114
views
Poincare inequality on the hemisphere
Background:
Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
- 537
2
votes
0
answers
78
views
Array-determined operator ideals
For a Banach space $X$, we, of course, know what it means for a sequence to be weakly null (to converge to zero in the weak topology).
An array in the Banach space $X$ is a sequence of sequences, $(...
- 121
2
votes
0
answers
319
views
What are alternative or equivalent definitions of a positive-definite function on a group?
The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
- 63
2
votes
0
answers
193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
2
votes
0
answers
45
views
Topology of an orbit space constructed from a Fréchet space under the "local" action of some "smooth" group
Let $G$ be a nontrivial connected compact subgroup of the general linear group $\operatorname{GL}(\mathbb{R}^3)$. For example, we may take $G$ to be $\operatorname{SO}(3)$.
Next, let $\mathcal{S}(\...
- 3,477
2
votes
0
answers
78
views
Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem
Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
- 809
2
votes
0
answers
103
views
Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
- 90
2
votes
0
answers
139
views
Multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
How to rigorously demonstrate that multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
I am utilizing the exponential type definition established by Elias Stein in the book 'Fourier ...
- 21
2
votes
0
answers
78
views
Converse of existence of minimizers
Let $(V,\|\cdot\|)$ be a real normed linear space. $V$ has the property that given any nonempty convex, closed subset $K$, there exists a unique $v_0\in K$ such that $\|v_0\| \leq \|v\|, \forall v\in ...
- 131
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
2
votes
0
answers
172
views
AQFT from a Lagrangian
In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...
- 424
2
votes
0
answers
98
views
Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?
Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$.
By Helmholtz ...
- 3,477
2
votes
1
answer
196
views
Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
2
votes
0
answers
144
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
- 21
2
votes
0
answers
83
views
Closed form solutions to polynomial operator equations
To the best of my knowledge the problem at hand is a generalisation of monic matrix polynomials. Can a closed form solution to the following equation be found,
$$u_3A_3X^3B_3 + u_2A_2X^2B_2 + ...
2
votes
0
answers
177
views
What are the current open problems in dilation theory?
I just started doing my PhD in mathematics. My topic is unitary dilations of operators. I've been reading a lot of papers on that subject so far (especially about the dilation of $n \ge 3$ commuting ...
- 63
2
votes
0
answers
89
views
On dense subspaces of $L^p$-spaces of finitely additive measures
Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...
- 557
2
votes
0
answers
95
views
Self adjoint operators from energy functionals
It is known that the equation
$$
\Delta f = 0
$$
on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...
- 115
2
votes
0
answers
102
views
Orthogonal representation of free products of two groups
Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
- 301
2
votes
0
answers
89
views
How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$
Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
- 63
2
votes
0
answers
63
views
Continuous-time Wold decomposition
I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line.
I am aware of the classic result in the book from Rozanov, which ...
- 21
2
votes
0
answers
73
views
Derivative of a functional involving integral and level set
Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional
$$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$
where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
- 29
2
votes
0
answers
164
views
$H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)
This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao.
There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
- 3,477
2
votes
0
answers
119
views
Limit of a distribution using Hörmander’s theorem
Let $\alpha \in \mathbb{C}$. I want to prove that
$$ (e^{i2\theta}\xi_1^2 + \xi_2^2 + \dots + \xi_n^2)^{-\alpha} \longrightarrow (Q(\xi)-i0)^{-\alpha}, $$
in $D’(\mathbb{R}^n\setminus \left\{0\right\})...
- 319
2
votes
0
answers
98
views
Geometric interpretation of uniform convexity condition
I first want to recall the moduli of uniform smoothness (US), uniform convexity (UC), asymptotic uniform smoothness (AUS), and asymptotic uniform convexity (AUC). Throughout, let $X$ be an infinite ...
- 41
2
votes
0
answers
170
views
finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
- 4,115
2
votes
0
answers
120
views
Closure of Laplacian
Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator
$$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$
There are two ...
- 1,171
2
votes
0
answers
62
views
Localized estimate for divergence free vector field
Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
- 31
2
votes
0
answers
188
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
- 1,171
2
votes
0
answers
320
views
Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable
I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
2
votes
0
answers
329
views
Conditions for an existence of smooth solution to a parabolic PDE
I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset):
\begin{equation*}
u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
- 425
2
votes
0
answers
157
views
Why do von Neumann algebras possess identity?
My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
- 433
2
votes
0
answers
180
views
Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
2
votes
0
answers
78
views
Analogy between quasi-injective modules & extensible Banach spaces
Let $X$ be a module. $X$ is said to be quasi-injective if every homomorphism $h:A\to X$ from any submodule $A\subseteq X$ has an extension to an endomorphism $\tilde{h}:X\to X$.
A module $X$ is quasi-...
- 2,605