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

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58 views

Dominated convergence theorem and fundamental lemma [migrated]

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
2
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0answers
102 views

Heat semigroup estimate on complete Riemannian manifold

Consider a complete noncompact Riemannian manifold $M$ such that the heat kernel $h_t(x, y)$ satisfies $h_t(x, y) \leq Ct^{-n/2}$. Consider a function $u \in L^p(M)$. How can we prove that ...
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1answer
117 views

Does the countable $\sigma$-product of a separable Hilbert space have a first countable topology?

Let $\mathcal{l}^2$ be "the" separable real infinite dimensional hilbert space, e.g. the space of square-summable sequences of real numbers. Let $\Box^{\mathbb{N}}\mathcal{l}^2$ be the countable ...
3
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1answer
141 views

Sobolev spaces on compact manifolds

Let us consider a self-adjoint elliptic pseudodifferential operator $P \in OPS^2$ on a compact manifold $M$ such that $spec(P) \subset (0, \infty)$. Is the norm $(Pu, u)^{1/2}$ on $H^1(M)$ equivalent ...
3
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1answer
65 views

Domain of square root of a self-adjoint positive operator [closed]

Let $A \geq 0$ be a densely defined self-adjoint positive operator on a Hilbert space $H$ obtained by Friedrichs extension, and let $Q$ be the densely defined quadratic form associated to $A$, that ...
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25 views

Fractional Laplacian on the torus [migrated]

Consider the Laplacian on the $n$ dimensional torus $T$, given by $-\Delta : L^2 \rightarrow L^2$. Let the domain of $-\Delta$ be all $C^\infty$ functions initially. Now consider the Friedrichs ...
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74 views

Boundedness of a Hilbert space projection map

Reading this recent thread I was reminded of a related problem I still haven't solved so I post it here in hopes of a positive result. Let $V_0 \subset H_0$ and $V_1 \subset H_1$ be separable ...
3
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1answer
118 views

Basis equivalent with a monotone basis

Given a basis in a Banach space $X$, can one find, for every $\varepsilon>0$, an equivalent basis with basis constant at most $1+\varepsilon$? In $L_p[0,1]$ with $1<p<\infty$ any monotone ...
3
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0answers
51 views

On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm $$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m ...
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0answers
41 views

persistence of regularity for nonlinear Klein-Gordon equation

I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" ...
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58 views

Carleman Estimates in a Riemannian manifold

Suppose $(\Omega,g)$ is a Riemannian manifold where $\Omega$ is a domain in $\mathbb{R}^3$ with smooth boundary. Furthermore suppose there exists a global coordinate chart $(x_i)$ such that $g= dx_1^2 ...
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0answers
95 views

Transitive closure of balanced bounded mass transport

Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
5
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1answer
113 views

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, ...
2
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1answer
213 views

Existence of a projection operator onto subspace of Hilbert space

Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is not ...
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0answers
59 views

About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...
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62 views

Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve $$\int_0^\infty\int_\Omega \nabla v\nabla ...
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0answers
31 views

Uniform bound in Faedo-Galerkin method with time-dependent weight in inner product

Let $v_j$ be an orthonormal basis for $V=H^1(\Omega) \subset L^2(\Omega)$ which is orthogonal in $L^2(\Omega)$. Let $w:[0,T]\times\Omega \to \mathbb{R}$ be a time-dependent weight which is smoooth ...
6
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0answers
140 views

Max min of functionals

I have an interesting question which I believe was probably already studied, but I could not find anything. Let $n, m \geq 1$ be fixed. Suppose that $|| \cdot ||$ is a norm in $\mathbb{R}^n$ and $f_1, ...
6
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2answers
321 views

Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...
2
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0answers
118 views

Weak topology on subsets of a normed space

I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset. When is the norm a continuous function on $E$? When is the metric induced by the ...
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0answers
40 views

Approximation property of Fréchet if range is restricted to an embedded Hilbert space

Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...
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1answer
100 views

Weak convergence in $W^{1,p}_0$

Note from the answerer : this question stems from this article. I ask this question in http://math.stackexchange.com/questions/1206617 I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so ...
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1answer
185 views

$\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?

If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms $$\mathcal N_p (\varphi)= \sum_{|\alpha|, |\beta| \leq p} ...
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71 views

absolutely continuous of two probability measures

Suppose $X_t$ satisfies $$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...
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1answer
82 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...
1
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1answer
141 views

How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...
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15 views

Approximate point spectrum is complement of set of points of regular type [migrated]

I have a question concerning the approximate point spectrum of a closed linear operator. I need to show that the approximate point spectrum is the complement of the set of points of regular type, ...
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32 views

Averages of bounded function [migrated]

For a continuous $f\colon\mathbb{C}\longrightarrow[0,1]$, what could be said about $\mathcal{F}=\{f_r\colon r>0\}$ where $$\forall\,r>0,\,\forall\,z\!\in\!\mathbb{C},\quad ...
1
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1answer
109 views

Bounded-open topology vs norm on $L\left(X,Y\right)$

In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform ...
4
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3answers
521 views

reflexive banach space

I want to ask this non-expert question: What does it mean geometrically for a Banach space to be reflexive? Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...
3
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2answers
190 views

distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...
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0answers
109 views

Dual space of $l^p(\mathbb{Z},X)$

Let $X$ be a Banach space, $p \in [1,\infty)$ and $l^p(\mathbb{Z},X)$ the usual sequence space taking values in $X$. Is it always true that $(l^p(\mathbb{Z},X))^* = l^q(\mathbb{Z},X^*)$ and ...
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1answer
135 views

Analysis of Sobolev spaces [closed]

I just wanted to know wthether the following is OK or not. Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
2
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0answers
98 views

Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
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76 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...
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0answers
66 views

Da Prato's notion of Symmetric Operator

For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ...
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0answers
104 views

Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on $\mathbb{R}^{+}$): $$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$ We call ...
3
votes
1answer
55 views

Relatively compact sets in Ky Fan metric space

Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E ...
3
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0answers
66 views

Is Wiener's Tauberian theorem true in Wiener space?

Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$. Is the following true? ...
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168 views

Lipschitz-free spaces of $\mathbb R^n$

We define $$ \text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and } \sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty. \} $$ It is well-known ...
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92 views

Topological properties of space of Radon measures

Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...
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1answer
84 views

About the upper bound on the roots of the matching polynomial

Heilman and Lieb had proven that if a graph had $d$ as its maximum vertex degree then the roots of the matching polynomial are bounded from above by $2\sqrt{d-1}$. Is there a modern exposition of ...
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1answer
89 views

Orthogonal compact operators on an infinite dimensional Hilbert space [closed]

How do I show that when $H$ is an infinite-dimensional Hilbert space we can find two compact positive operators $u,v$ with infinite dimensional image and $u \perp v$? This statement can be found at ...
2
votes
2answers
97 views

Infinite direct sum of $l_2^{(n_k)}$ contains a complemented isometric copy of $l_2$

How do I show that for any increasing sequence $(n_k) \subseteq \mathbb{N}$, the space $\left( \oplus _{k=1} ^\infty l_2 ^{(n_k)} \right) _\infty$ contains a complemented isometric copy of $l_2$?
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1answer
109 views

Is a Fréchet Montel space distinguished?

Based on a couple of references, it seems that the answer is yes, see for example Boneta-Dierolf, 1992 and Bierstedt-Bonet, 1989. However, from a comment to the answer of this MO question, I infer ...
5
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85 views

A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
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86 views

Banach space dual to $L^\infty(I,H^1(M))$

What is the dual to $L^\infty (I,H^1(M))$?, where $I$ is an interval in the real line; $H^1(M)$ is Sobolev space of degree 1, and $M$ is a compact manifold like the torus. Any references that show ...
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28 views

Lipschitz continuity of a composition operator

Let $P\colon U\subset \mathbb{R}^K\rightarrow \mathbb{R}^K$ be $C^\infty$, $\Omega \subset \mathbb{R}^d$ open and bounded, $m \in \mathbb{N}$, $p\in [1,\infty]$ such that $m>\frac{d}{p}$, $f \in ...
4
votes
1answer
79 views

Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?: Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
0
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1answer
45 views

Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$. At a certain part in a proof I ...