# fa.functional-analysis's questions - English 1answer

5.222 fa.functional-analysis questions.

### 3 Is the compact-open topology on the dual of a separable Frechet space sequential?

Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...

### 10 Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

1 answers, 862 views fa.functional-analysis sobolev-spaces

### 4 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 ...

### 2 What restrictions on the form of an integral equation have a unique solution f=0?

We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...

### 1 small perturbation of transfer operator without discrete spetrum

Pommeville-Manneau maps: $T_{\alpha}=x+2^{\alpha}x^{\alpha+1} x \in [0,\frac{1}{2}], 2x-1, x \in [\frac{1}{2}, 1], \alpha <1$ is well known to have polynomial decay of correlation, it transfer ...

### 1 Bounding the norm of the Laplacian of the gradient of a function having Lipschitz continuous Hessian

It seems that the following claim is true, but I did not manage to prove it neither to find a reference. Claim Let $f:\mathbb R^p\to\mathbb R$ be a three times differentiable function such that its ...

### 2 $L^p$ distance of divergence-free vector field

Let $F_1, F_2$ be two divergence-free vector fields on a simply connected region $\Omega \subset R^3$, and suppose $|| F_2-F_1 ||_{L^2(\Omega)} \leq c|| |F_2| -|F_1| ||_{L^{\infty}(\Omega)}$, for ...

### 1 Reference Request: Differentiability of Moreau Envelope

I've recently come across many results discussing the differentiation of the Moreau envelope defined by $$e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z) ,$$ where $f$ is a convex functional on a ...

### 7 Fourier dimension of the sum of sets

This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...

### 1 Efficiently reversing the triangle inequality with additional information

2 answers, 312 views fa.functional-analysis real-analysis
Suppose $f$ and $g$ are bounded functions, having whatever niceness properties you want, on some space of finite measure. Assume they are normalized so that $\int |f|^2=\int|g|^2=1$. I am looking for ...

### 1 Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...