Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
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Covering rough boundaries of closed sets in manifolds by charts
This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible.
Consider a Riemannian ...
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Negative eigenvalue of Toeplitz Hermitian matrix?
I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
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extension for a complex operator
Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
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The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
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Lower bound for the $C^*$-unitisation norm?
Let $A$ be a $C^*$-algebra without unit. Its unitisation $A^+$ carries the $C^*$-norm
$$\|(a,\lambda)\|_{A^+} = \sup \{\|ax+\lambda x\|_A \;\big|\; \|x\|_A = 1 \},$$
which is the operator norm of $...
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$C^{1,2}$ regularity of (weak) solutions to the heat equation
Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation
$$u_t - \Delta u = 0$$
$$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$
$$u(0) = u_0$$...
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A problem on Markov chains and Dirichlet forms
Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy
$$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$
$$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$
$$c(x,x)=0\text{ for ...
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How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?
Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$.
Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\...
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Multilinear Interpolation
Suppose I have a multilinear map $\Gamma(u,v)$ satisfying
\begin{align}
\big\| \Gamma(u,v)\big\|_{L^2} &\leq \big\| u\big\|_{L^2} \big\| v\big\|_{L^2} \\
\big\| \Gamma(u,v)\big\|_{L^\infty} &\...
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Boundary regularity of higher order PDE
consider the subsequent pde (weak formulation):
$\int_\Omega D^m\phi:D^m\psi+ Df(D\phi):D\psi+(g h\circ\phi)\cdot\psi dx=0$.
In this case, $n\geq 2$, $\Omega=[0,1]^n$, $m>2+\frac{n}{2}$,
$\phi\...
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Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space
I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces".
We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, ...
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Finding the infimum of the range of a certain non-negative function associated to a $ C^{*} $-algebra
Let $ A $ be a non-trivial $ C^{*} $-algebra and $ n \in \mathbb{N} $. Setting $ \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} $, we can define a function $ f: \...
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inverse problem to resolution of the identity
Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\...
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Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
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Alternative representation of $C_c(X)$ as inductive limit
CORRECTION: As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear.
Under some additional constraints on the space (e.g. $X$ ...
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About hyperplane separation theorem
I read in Lovasz's notes about semidefinite programs and combinatoric optimization.
If $x_1A_1 + ... + x_nA_n\succ 0$ has no solution, then the linear subspace $L = x_1A_1 + ... + x_nA_n$ is ...
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Boundedness of a function that satisfies a PDE-type inequality
Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$.
Suppose that
$$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + \int_{...
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How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?
Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of
$$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$
$$\partial_\nu u = \alpha \quad \text{on $\...
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Weak convergence in $L^2(0,T;X)$
In the book Navier Stokes Equations by Constantin and Foias, the folloiwng argument is made:
Let $u_m$ converges weakly to $u$ in $L^2(0,T;V)$ where
$$
V=\overline{\{f\in (C_0^\infty(\Omega))^...
user14319
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Does bounded and closed equal compact for sets of Borel probability measures?
Equip the space of Borel probability measures on a fixed closed subset X of the s-dimensional Euclidean space with the topology induced by weak convergence of probability measures. In this setting, ...
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Can Gradient be controlled by Curl and Divergence in Morrey spaces
In $L^p(\mathbb{R}^3)$, it holds for $1< p< \infty$ and $\mu\in C^\infty_0(\mathbb{R}^3)$,
$$\|\nabla\mu\|_p\leq C \left( \|\operatorname{div} \mu\|_p + \|\nabla\times\mu\|_p \right).$$
So, how ...
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The real method of interpolation and operator ideals
Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, ...
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Partial trace with spectral measure
I'm a physicist who needs mathematical advice:
Let $A= \sum_{i=1}^{\infty} a_i P_{\phi_i}$ be a self-adjoint operator with projectors $P_{\phi_i}$ on the orthonormal eigenbasis $(\phi_i).$ Let $$B= \...
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Heat semigroup ultracontractive?
Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
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On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $
This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) $...
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The closure of a set of closed points
Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the ...
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Asymptotics of a elliptic pde when exponent gets large
I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
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One-parameter group of unitary operators and Core
Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
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Amenable hypergroups
I needed some information about the intersection of the kernels of invariant means on hypergroups. So I read the discussion made for the question " The kernel of all invariant means " which answer my ...
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A Question about compactness of an embedding into $L^p$ spaces
Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} \frac{u^...
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Zeros of functions constituting a Riesz-basis for the Paley-Wiener space
I have a short question which first requires some slightly elaborate definitions:
Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...
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When is an injective Fredholm map on $\ell^p$ a diffeomorphism
Consider a family of maps $f_p\colon \ell^p(\mathbb Z,\mathbb R)\to \ell^p(\mathbb Z,\mathbb R)$, $p\ge 2$, where
$$
f_q = f_p\big|_{\ell^q},\qquad \forall\; 2\le q\le p.
$$
Moreover,
$f_p\colon \...
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The dual of the space of smooth functions that vanish at infinity
Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...
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In which sense Daubechies wavelets converge to the Shannon wavelet?
My question is about wavelets theory. Consider $\psi_n$ the Daubechies wavelet of order $n \geq 1$; that is, the Daubechies wavelet with $n$ vanishing moments. We also define the Shannon wavelet in ...
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How to define Biharmonic operator for second order sobolev spaces
I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have
$$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : W^...
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Homomorphic Commutator? Equation
So I was considering the following functional equation:
Given $H :\Bbb{C}^2 \rightarrow \Bbb{C} $ find $\theta: \Bbb{C}^2 \rightarrow \Bbb{C}$ such that
$$ \theta(H(a,b), H(c,d)) = H(\theta(a,c), ...
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Comparing Dirichlet energy and area of a Surface-immersion
Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
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Construct a PDE solution from a net of approximations
Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let $...
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Weighted Sobolev spaces over open/closed intervals
I am escalating this question from SE, as I have been unable to obtain any guidence in relation to a possible solution.
Some context, I am working with weighted Sobolev spaces of the form $W^{m,2}(I,...
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Density of subspace with nonlocal/Wentzell boundary condition
Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by $E=\...
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Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
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Regularity of weak solution
I have also posted the question here. Let me explain what difficulties I have. In fact, one may write
\begin{equation}
\partial_1(f-\partial_1 u)=0
\end{equation}
in $\Omega$. Then one may have the ...
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One-sided local $L^p$ spaces
Consider the vector space $L^p_{\text{left-loc}}$ of measurable functions $f:[0,1]\to\mathbb R$ so that for all $x\in(0,1]$ there exists $\delta>0$ so that $f|_{[x-\delta,x]}\in L^p$.
Does this ...
- 8,086
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Cuntz comparison of strictly positive elements in finite C*-algebras
Let $A$ be a finite, non-unital C*-algebra, $s\in A$ a strictly positive element and $a\in A$ a positive element that is Cuntz-equivalent to $s$, i.e. there exist sequences $\{x_n\},\{y_n\}\subset A$ ...
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Condition for maximizer of convex combination to be expansion mapping
I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$
$$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$
such ...
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pettis integral
Maybe this is rather a refernce question on Pettis integrals. Some naive questions arise:
1) Assume that $F$ is Pettis-integrable on $\Omega$ and that $\omega \subset \Omega$ is measurable. Is $f$ ...
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Vector valued Sobolev spaces
My question is in reference to this question previously asked here. As asked there, consider a function $f \in H^s_x(L^2_y) \cap L^2_x(H^s_y)$. In the notation of Lions and Magenes (Chapter 4, Vol 2), ...
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Counting variables to look for invariances/range conditions
A while back, I asked this question on m.se. I wasn't terribly happy with the answer, and when someone asked a very similar question which isn't getting any action, it got me thinking again. Let me ...
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Sobolev embedding on warped product
Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote ...
user73978
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Related to derivative of Modified Bessel I function wrt the order
I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that $Re(\dfrac{I'...
