All Questions
Tagged with fa.functional-analysis reference-request
1,021 questions
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 463
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votes
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66
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Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 463
1
vote
0
answers
61
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Primes as the extrema of a functional
I'd like to write down a functional on sequences for which the prime numbers are an extrema.
One generally thinks of the natural numbers first as an ordered set, and then you discover unique ...
- 571
2
votes
0
answers
155
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Second differential of total variation
I am trying to give meaning to the notion of second differential of total variation.
For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by
$$TV(u)=...
- 6,367
7
votes
2
answers
641
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Decay of solutions to Schrodinger equation with local minimum in potential
Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, and ...
- 9,007
2
votes
2
answers
230
views
Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation?
Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu_f, \mu_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (...
- 12.9k
3
votes
2
answers
808
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Growth of $L^p$ norms as $p \to \infty$
Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
- 128k
2
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0
answers
149
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Reference for weighted Sobolev spaces
I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
- 4,713
2
votes
0
answers
161
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The Laplace transform and the Lagrange compositional inversion formula
I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
- 10.5k
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votes
0
answers
59
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Identification of vector valued function
Do you know of a good reference for a proof of the fact that $L^2(0,T,L^2(\Omega))$ and to $L^2([0,T]\times \Omega)$ can be identified?
- 235
0
votes
1
answer
93
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References and updates on a $L^p$ Factorization theorem by Maurey
In the "Exposé XV" of the 1972-1973 of the Maurey-Schwartz Seminar of functional analysis ("Théorèmes de factorisation pour les opérateurs linéaires à
valeurs dans un espace $L^p(\Omega,...
- 31.5k
3
votes
0
answers
190
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$C^1$-regularity of solution of a Dirichlet problem
I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
- 2,494
5
votes
1
answer
224
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Spectral theory of infinite volume hyperbolic manifolds
I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
- 1,713
1
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0
answers
110
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Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
4
votes
1
answer
281
views
Does property (V) imply the Grothendieck property for dual Banach spaces?
A Banach space $X$ has property (V) whenever for each Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact; equivalently, every non-weakly compact operator $T:X\to ...
- 2,605
3
votes
0
answers
108
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On the derivatives of the solutions of the heat equations with Neumann boundary condition
Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
- 721
2
votes
1
answer
246
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Change of variable and boundary data for Laplace equation
Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem
$$
\begin{cases}
-\Delta u = 0 & x \in \Omega \\
u = 1 & x \in \partial \Omega
\end{cases}
$$
Does it make ...
- 1,422
1
vote
1
answer
452
views
Uniformly convex Banach spaces
Theorem. If $X$ and $Y$ are uniformly convex Banach spaces, then for $1<p<\infty$ the space
$$
X\oplus_pY=X\times Y,
\qquad
\Vert(x,y)\Vert:=(\Vert x\Vert_X^p+\Vert y\Vert_Y^p)^{1/p}
$$
is ...
- 28k
2
votes
1
answer
247
views
Is $L^p(X,\ell^p)$, $1<p<\infty$, uniformly convex?
Let $(X,\mu)$ be a measure space and let $1<p<\infty$.
Question. Is the space $L^p(X,\ell^p)$,
$$
\lVert f\rVert_p=\Bigl(\int_X\sum_{i=1}^\infty \lvert f_i\rvert^p\, dx\Bigr)^{1/p},
\qquad
f=(...
- 6,035
3
votes
1
answer
210
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Reference on inductive (direct) limit of ordered vector spaces and vector lattices
I looked in all textbooks on vector lattices (Riesz spaces) as well as ordered vector spaces, but couldn't find any mentions of neither inductive nor projective limit for these structures. Googling ...
- 46
4
votes
1
answer
155
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Resource on spectral theory for differential operators with symmetry groups
In Methods of Mathematical Physics IV by Reed and Simon, the authors cover Floquet theory in detail in Section XIII.16. On page 280, they note that
"A part of the analysis of [the periodic ...
- 133
13
votes
2
answers
915
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Topological vector spaces (reference request)
In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
- 3,738
4
votes
0
answers
164
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Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
2
votes
0
answers
110
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About Hilbert-Haar theory
The Hilbert-Haar theory says that functionals $\mathcal{F}(u,B)=\int_{B} F(\nabla u)\,dx$, where $F$ is a convex function and $B$ is a bounded domain in $\mathbb{R}^N$, take a minimum in the space of ...
- 63.9k
1
vote
0
answers
142
views
Uniformly continuous semigroups are analytic
Reposting from stackexchange.
I know that every analytic $C_0$-semigroup is differentiable and then every differentiable semigroup is norm continuous.
I want to know where uniform continuity fits in ...
- 6,367
1
vote
1
answer
250
views
Moser/Schauder estimates for coercive boundary conditions
Consider the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on $(0, \infty) \times \Omega$, where $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary, and $L$ is a ...
CommunityBot
- 1
4
votes
2
answers
428
views
Are sequences in $\ell^1(\mathbb N_0)$ converging uniformly on convex weakly compact subsets of $c_0(\mathbb N_0)$ norm convergent?
I think the question as expressed in the title should be clear. I do not know whether there is a known "characterization" of the weakly compact convex sets in $c_0(\mathbb N_0)$ but testing ...
- 2,605
13
votes
1
answer
3k
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Does this metric have an official name? Lévy metric? Ky Fan metric?
Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...
- 111
6
votes
2
answers
240
views
Continuity of a differential of a Banach-valued holomorphic map
Originally posted on MSE.
Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
- 3,738
10
votes
2
answers
594
views
Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
- 18.7k
3
votes
0
answers
158
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$L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian
Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
- 1,407
3
votes
1
answer
159
views
Tight L2 bound on moments approximation and reference
Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$.
The error in approximated the moments ...
- 18.7k
3
votes
2
answers
352
views
Reference for commutator estimate
I'm interested in Sobolev space estimates for commutators involving a pseudodifferential operator and a Fourier multiplier. More specifically, suppose $p = p(x,\xi) \in S_{1,0}^{m_1}$ and let $q = q(\...
- 138
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0
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540
views
The definition of essential spectrum for general closed operators
I've asked this problem in MSE several days ago, see here. But there is no reply up until now. Maybe I wrote things too complicated there and so I'll write a very clean problem here. For background ...
- 1
3
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0
answers
479
views
de Rham currents/distributions on manifolds with boundaries
My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
3
votes
0
answers
170
views
A version of the Nash-Moser inverse function for unbounded domains?
Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $\mathbb{R}^n$? Any reference would be appreciated but ...
- 505
4
votes
1
answer
221
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Fourier multipliers and transference on cyclic groups
It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\...
4
votes
0
answers
160
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An estimate for the Benjamin-Ono equation from T. Tao's well-posedness paper
In https://arxiv.org/abs/math/0307289 (eq. (8)),
for a (smooth) solution of the equation $$u_t - uu_x + Hu_{xx} = 0$$
(where $H$ denotes the Hilbert transform) the following estimate is stated (...
- 839
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votes
2
answers
2k
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What is the "Krein-Milman theorem for cones"?
Update: The question is completely answered. I had overlooked a reduction to the self-adjoint case, and the latter can be proved using a Hahn-Banach separation theorem. Thanks to Matthew Daws for ...
- 63.9k
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0
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79
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Reference for smoothness of Nemytskii operator on fractional Sobolev spaces
Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator
$$
\big(N_\varphi x\big)(t)=\varphi\big(x(t)\big)
$$
for $x\in H^s(T^d)$, the ...
- 93
0
votes
0
answers
70
views
Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99
2
votes
1
answer
399
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Smoothness of Minkowski functional is equivalent to smoothness of boundary
Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$,
$$
f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\},
$$
is $C^1$ ...
- 63.9k
7
votes
3
answers
2k
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Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs
What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in
real analysis,
complex analysis,
functional analysis,
ODEs,
PDEs?
The ...
- 6,367
1
vote
1
answer
197
views
Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous
This is followup to the following question: On the Lipschitz continuity of the unit-normal vector field of a polytope
Let $C$ be a (nonempty) closed convex subset of $\mathbb R^n$. Note that to every ...
- 7,633
3
votes
1
answer
178
views
Looking for a paper on axiomatic orthogonality in a vector space
I am looking for a paper "Linear spaces with disjoint elements and their conversion into vector lattices" by A. I. Veksler.
It was published in 1967 in Research Notes of Leningrad State ...
- 37.8k
2
votes
1
answer
432
views
Second-order elliptic regularity with rough coefficients
Let $\Omega \subseteq \mathbb R^n$ be a bounded open set with smooth boundary. Let $k\geq 1$, $\alpha\in(0,1)$, $a_{ij},b_i,f \in C^{k,\alpha}(\Omega)$ for $i,j=1,...,n$, and define the operator
$$L = ...
- 2,494
2
votes
0
answers
559
views
Multiplication in Sobolev space with negative exponent
My initial problem is the following: I would like to estimate $\lVert f^2\rVert_{H^{-2}}$ in the sense of Sobolev embeddings, where $f:\Omega \rightarrow \mathbb{R}$ is a function defined on a bounded ...
- 914
10
votes
1
answer
593
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Density of smooth function in Hilbert spaces
I am looking for a simple reference to the following fact:
If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\...
- 63.9k
1
vote
1
answer
277
views
Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$
Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets
$$\begin{align*}
S_1 &= \left\{ \begin{...
CommunityBot
- 1
1
vote
0
answers
56
views
Smooth approximation in Sobolev spaces for surfaces with boundary
Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
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