0
votes
1answer
63 views

reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$

I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
6
votes
2answers
179 views

Literature on “real” $C^*$-algebras

I am trying to get a better understanding of "real" $C^*$-algebras. I encountered them in the paper D. Voiculescu, Dual algebraic structures, J. Operator Theory 17(1987), 85-98, which cites G.G. ...
0
votes
1answer
78 views

Fractional Laplacian on compact hypersurface/manifold via harmonic extension?

Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$. In pages 10-11 of this paper, the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the ...
0
votes
1answer
85 views

Nonlocal (parabolic) PDEs in the Sobolev space setting

Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form $$u_t + (-\Delta)^s u = f$$ where the nonlocal operator is the fractional Laplacian) in the setting of Sobolev ...
1
vote
0answers
153 views

Smoothness of eigenfunctions of integral equation

Can you provide a proof or a reference, to study from, for the following problem: Assume $\Gamma$ is a real analytic closed rectifiable curve in the plane, $ds$ is the arc-length and kernel ...
3
votes
1answer
285 views

A useful criterion in vector integration

I would like to know the proof of the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let ...
0
votes
0answers
92 views

A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form $$u_t(t) - ...
1
vote
1answer
76 views

L logL space and compactness

I think that if a sequence of L^1 functions have the integral $$ \int f_n \log (f_n)dx $$ uniformly bounded, then there is a subsequence that converges strongly in $L^1$. The questions are: 1) Is ...
2
votes
1answer
92 views

Reproducing kernels and equivalent inner products

Suppose $H$ is a reproducing kernel Hilbert space and $K_{1}\left(x,\cdot\right)$ and $K_{2}\left(x,\cdot\right)$ two reproducing kernels with respect to two equivalent inner products on this space. ...
10
votes
2answers
299 views

Reference for invariance of essential spectrum under relatively compact perturbations

I'm looking for a proof of the following statement: Let $X$ be a Banach space and $T$ be a closed map on $X$. For any relatively compact map $A$ the essential spectrum of $T$ and $T+A$ are the same. ...
1
vote
1answer
58 views

Reflexive Besov spaces Bs,p,q

I don't know whether the Besov space $B^s_{p,q}$ with $1<p,q<\infty$ is reflexive or not? Can someone help me please?
5
votes
1answer
188 views

Self-adjoint operator

Assume that $B$ is a self-adjoint operator and $\alpha\in(0,1)$. I need a reference for the following equality $$B^{-\alpha}=\frac{\sin\alpha \pi}{\pi}\int_0^\infty ...
2
votes
0answers
117 views

Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets

I was involved into this subject when I answered this question from MSE. Trying to generalize my answer, I am thinking about a following Question. Let $X$ and $Y$ be metric spaces. When each ...
3
votes
0answers
139 views

Exterior powers and singular values on Hilbert spaces

I am currently writing an article relating to multiplicative ergodic theorems for cocycles of bounded operators acting on a Hilbert space, and in parts of the argument it is necessary for me to refer ...
4
votes
0answers
155 views

On the classification of injective Banach spaces

A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a ...
0
votes
2answers
130 views

A basic question about JL Lions' transformation of a Stefan problem

In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The ...
4
votes
2answers
233 views

Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces

This question was posed on MathStackExchange but did not get an answer (even with a bounty). In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert ...
2
votes
1answer
71 views

Support-preserving pseudodifferential operators

Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the ...
0
votes
1answer
170 views

Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous? I define $H^{\frac 1 ...
0
votes
2answers
71 views

References for well-posedness of weak solutions to Stefan problem

Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)? ...
0
votes
2answers
117 views

Defining surface integral on boundary of $C^1$-domain

Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to ...
5
votes
1answer
97 views

A question about extensions of Markov semigroups

I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. ...
3
votes
0answers
87 views

Any references on infinite-dimensional Fourier-Plancherel theory?

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...
1
vote
1answer
112 views

If $u \in W^1(0,T;L^2,H^1)$ and $\varphi \in C^1([0,T]\times \Omega)$ then $\varphi u \in W^1(0,T;L^2,H^1)$?

Let $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Define $$W^1 := W^1(0,T;L^2,H^1) := \{w \in L^2(0,T;H^1(\Omega)) \mid w' \in L^2(0,T;H^{-1}(\Omega))\}$$ where $w'$ means the weak ...
1
vote
2answers
190 views

Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{|\cdot|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
5
votes
1answer
191 views

definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the transformation $ (T − \lambda)/(T + \bar \lambda\ ) $ with domain and range in the Hilbert space is contractive for ...
7
votes
2answers
850 views

Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
7
votes
1answer
208 views

Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
2
votes
1answer
94 views

Linear combination of i.i.d. $Z_i$ distributed as $Z_1$

A classical property of the Gaussian distribution is that, if $\{Z_i\}_{1 \leq i \leq n}$ are i.i.d. standardised Gaussian distributions (i.e. $Z_i \sim N(0,1)$) and $S = \sum_{i=1}^n a_i Z_i$ where ...
1
vote
1answer
130 views

Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
2
votes
0answers
68 views

Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in ...
3
votes
1answer
205 views

Orthonormal basis in $\ell^n_p$

Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of $$ \sum_{i=1}^k \|a_i\|_{\ell^p}^2 $$ for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis. ...
7
votes
2answers
273 views

Measures whose projections are absolutely continuous

Since my question was not answered on MSE, I would like to ask it here. Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...
3
votes
0answers
215 views

Dual of $L^2(0,T,C)$ where $C'=BD(\Omega)$

Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary. I'm actually looking for a proof that explains what is the dual of ...
1
vote
0answers
154 views

Composite families of formal power series over $\mathbb C$ as algebraic variety

I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
1
vote
1answer
129 views

Nonlinear parabolic PDEs existence with Galerkin method?

Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but ...
4
votes
0answers
194 views

Is there an “exponential law” for differentiable maps between smooth manifolds?

Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question: Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds ...
12
votes
1answer
323 views

Applications of non-separable Hilbert spaces

In applications, Hilbert spaces of interest are often assumed to be separable. In addition to being extremely convenient mathematically, this assumption can often be justified on computational or ...
1
vote
0answers
202 views

Is an exact operator, unitary equivalent to a banded operator?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is ...
9
votes
0answers
300 views

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 ...
1
vote
1answer
205 views

Reference request: Spectral analysis of advection diffusion PDE

As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and NOT on existence/uniqueness etc. which is usually the ...
1
vote
2answers
221 views

Is there a reference for compact imbedding theory of Hölder space?

This question is posted and unanswered from math.stackexchange. Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to ...
2
votes
2answers
174 views

Resource on Infinite Systems of Difference Equations

I have asked this question previously at Math.stackexchange, but it seems to receive little attention there. In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer ...
6
votes
1answer
191 views

Origin of the term Riesz Basis

The term Riesz basis is in wide use for the image of an orthonormal basis in a Hilbert space under a bounded invertible map (there are lots of equivalent definitions). But I've been unable to find ...
4
votes
1answer
214 views

Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
3
votes
0answers
58 views

$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology

This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
1
vote
1answer
163 views

If a function is defined in terms of a solution to an initial value problem, is it also solution to an initial value problem?

Say $f:\mathbb R^{n+1}\to \mathbb R^p$ is a solution to an initial value problem, and $g:\mathbb R^{n+1}\to \mathbb R^q$, so that the components of $g$ can be expressed as polynomials in $f$, $f'$, ...
4
votes
0answers
87 views

How fast is discrete-time diffusion on a continuous set?

This question is inspired by Joseph O'Rourke's beautiful answer to my previous question. Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
1
vote
1answer
121 views

Special kind of operators

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$ where ...
4
votes
0answers
198 views

When separation in $L^1$ is possible?

Let $A$, $B$ be disjoint convex closed subsets of the Banach space $L^1[0,1]$. Assume additionally that $A$ is bounded and $A$, $B$ are closed under convergence in measure. Then there exists a closed ...