Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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1answer
86 views

Various limits of the Christoffel Darboux Kernel

In a different thread, we stumbled upon the following question: Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...
6
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143 views

intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
8
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2answers
382 views

Attempted Banachification of a space

In a recent blog post, Terry Tao mentioned the question of how to tell if a Hausdorff topological vector space admits a finer topological structure which happens to be the topology of a Banach space (...
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61 views

Functions $f \in S(\mathbb{R})$ with slow decay rate?

I came across this question by thinking about whether there are functions $f \in S(\mathbb{R})$ that satisfy for all(!) $a>0$ that $f e^{a|.|^a} \notin L^{\infty},$ as somebody on math....
2
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1answer
206 views

Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\...
2
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1answer
180 views

Strong maximum principle for heat equation. Positivity of solution

I have a non-negative solution $u \in L^2(0,T;H^1) \cap H^1(0,T;(H^1)')$ of the heat equation $$u_t-\Delta u =0$$ on bounded $C^1$ domain $\Omega$, with the boundary condition $$\frac{\partial u(t,x)}{...
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57 views

On the set of times such that $e^{-it\Delta}f\not\in L^6(\mathbb{R}^3)$

Let $e^{-it\Delta}$ the flow of the free Schrodinger equation on $L^2(\mathbb{R^3})$. The (endpoint) Strichartz estimates implies that, given $f\in L^2(\mathbb{R^3})$, $e^{-it\Delta}f\in L^6(\mathbb{R}...
2
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1answer
188 views

Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
2
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1answer
111 views

Does weak convergence implies weak convergence of the positive part?

Weak convergence of a function in Sobolev spaces implies weak convergence of positive and negative parts of that function or not?
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218 views

Error estimate in the spectral theorem of compact operators on a Hilbert space

Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
2
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1answer
86 views

About eigen-functions of the Gaussian kernel

If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
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1answer
761 views

Two questions on Elias Stein paper (1976)

I am working on some results related with a paper of Elias Stein (on the almost every where convergence of wavelet summation methods), and I have the following questions: The maximal function ...
4
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101 views

Good reference for noncommutative $L^p$ spaces

I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
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110 views

Automorphisms of Cuntz algebra

Suppose, $ O_{\infty} $ is the cuntz algebra generated by the orthogonal isometries $ \{S_i\}_{i\in \mathbb{N}} $,i.e. $ S_i^*S_j=\delta_{ij}$ and $ O_{\infty}=C^*(\{S_i\}_{i\in \mathbb{N}}) $. Then ...
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117 views

Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
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55 views

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 ...
5
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1answer
113 views

$L^\infty$ estimate on heat equation with a lower order term

Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of $$u_t - \Delta u = f$$ $$u(0) = u_0$$ $$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$ ...
2
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1answer
86 views

Measurability of integrals with respect to different measures

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
4
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76 views

On the weakly sequential completeness of the dual of the James space $J$

Let me first introduce some definitions. Let $1\leq p\leq \infty$. A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is ...
4
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1answer
77 views

Operator space structures on CB(H,K) where H and K are Hilbertian operator spaces?

(I'd be grateful if anyone thinking of putting MathJax in the question title refrains from doing so.) By consulting various standard sources (Effros-Ruan's book, Pisier's book, the lexicon of ...
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2answers
311 views

$l^1$ versus $l^2$

Is there an elementary proof of this Banach space fact? If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite ...
2
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1answer
148 views

Simplify proof for rapidly decaying functions

I want to show the following theorem in a lecture: Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$ Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto (f_1(x),..,f_k(...
3
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0answers
60 views

A continuous functional calculus on/positive elements in a Fréchet algebra?

I am trying to understand what (minimal) conditions one would need in order to obtain a functional calculus on a Fréchet algebra, which we demand to be equipped with an involution that leaves all semi-...
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42 views

Can we say translation/dilation of the $L^p-$multiplier is again a $L^{p}-$multiplier?

Suppose that $m:\mathbb R \to \mathbb C$ such that $\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}$ (where $C$ is some constant, $f\in L^{p}$). (That is, $m$ is an $L^{p}-$ multiplier) ...
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42 views

1D inhomogeneous linear Schrodinger equation

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
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116 views

Average nastiness of a Newton polytope

Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity: $$ P(x)= \left( \sum_{\alpha \in P} \binom{d}{\alpha}...
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28 views

A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻² Let $f$ be ...
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1answer
51 views

A diagonalisation argument applied to density functions

There is a claim from a paper which I do not understand: Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})_{\eta >0}$ be a family of densities for random variables on $(C[0,T], \mathbb{R}^...
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1answer
262 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...
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67 views

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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161 views

Non-compact analogue of Peter-Weyl

I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L^2(G,\mu_H)$ decomposes as \begin{equation} \int^{\...
5
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1answer
162 views

Extension of functions from geodesically convex compact sets in a Riemannian manifold

In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
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77 views

Convolution Integral involving an unknown function

I've got the following problem I'm working on which is related to some of my research. I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \...
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57 views

What equals $\ker[(A-\lambda I)^+]$ for a negative unbounded operator $A$?

We have the following result: $\{ E_{\lambda}; \, -\infty <\lambda < + \infty\}$ is a spectral family, where $E_{\lambda}$ is the projection of $H$ onto the null space $\mathscr N \left(A_\...
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46 views

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} &\...
3
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1answer
157 views

Uniqueness from orthogonality relation?

This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...
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75 views

Getting an a priori energy estimate from PDE weak formulation

On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying $$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv$...
1
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1answer
217 views

On the second dual of $C[0,1]$

I have two questions on the second dual of $C[0,1]$: R. D. Mauldin ([1]) proved that: For a given bounded linear functional $T: C[0,1]^*\to \mathbb{C}$ there is a bounded function $\psi$ defined on ...
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3answers
124 views

Norm of an operator formed using a unitary operator

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
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90 views

Are Ritt operators mean ergodic?

In the following, $T$ is a bounded operator on a Banach space $X$. $T$ is called "power bounded" if $\sup_{n\in \mathbb N}\|T^n\|<\infty$; $T$ is called "mean ergodic" if the Cesàro sums $\frac{1}...
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113 views

The boundedness of an entire function along the imaginary axis

I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...
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31 views

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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5answers
503 views

Applications of functional analysis beyond analysis(towards algebra, geometry, number theory…) [closed]

So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional ...
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59 views

Topologies with the same convex closed sets

Let $\tau_1$ and $\tau_2$ be locally convex Hausdorff topologies on vector space $X$ such that $(X,\tau_1)^\ast = (X,\tau_2)^\ast$. It is well known that $(X,\tau_1)$ and $(X,\tau_2)$ have the same ...
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47 views

When does the ground state energy continuously depend on a parameter?

Given a family of Schrödinger operators $H_\gamma=-\Delta+V_\gamma$, under which condition is the map $\gamma\mapsto\inf\sigma(H_\gamma)$ continuous? This is surely the case for many textbook ...
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223 views

Is this continuous linear map weakly compact?

Let $E$ and $F$ be Fréchet spaces, let $U$ be an open subset of $E$, and let ${\mathcal{H}}(U;F)$ be the spaces of holomorphic mappings from $U$ into $F$. Let $\tau_c$ denote the compact-open ...
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0answers
57 views

Properties of convergence at points of continuity

Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps. Then $J$ is a ...
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3answers
497 views

is every element in a C* algebra a product of normal elements?

I have the following question and since I am not an expert on C*-algebras, I thought I ask it here: I know that in general the sum and product of normal elements need not be normal. It is even true ...
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1answer
54 views

Extremal of an L^1 continuous functional on a compact bounded set

Please, I need a small help with a reference. Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and ...
3
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0answers
140 views

A strongly open set which is not measurable in the weak operator topology

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$. Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...