fa.functional analysis – Confusing definition of homogeneous Sobolev norm of order -1

Let $Omega subset mathbb{R}^{d}$ and $|.|$ is the standard euclidean $2$-norm. I came across a definition of $dot{H}^{-1}(Omega)$ which is a bit confusing. In (1) authors define the following semi-norm for $hin C^{1}(Omega)$:
begin{equation}
|h|_{dot{H}^{1}}:=left(int_{Omega} |nabla h(x)|_2^{2} dx right)^{1/2}
end{equation}

where $dx$ writes for the Lebesgue measure. They define for a signed measure $nu$ on $Omega$ the norm $|nu|_{dot{H}^{-1}(lambda)}$ by standard duality arguments:
begin{equation}
|nu|_{dot{H}^{-1}}=sup_{|h|_{dot{H}^{1}}leq 1} |int_{Omega} h d nu|= sup_{|h|_{dot{H}^{1}}leq 1} |langle h,nurangle|
end{equation}

where I noted $langle h,nurangle=int_{Omega} h d nu$ which defines a inner product.
Authors argue that this Homogeneous Sobolev norm is finite for measure having zero total mass.

I was a bit confused: is this definition a particular case of the more “standard” one using tempered distributions and Fourier transform (see Definition 1.31 in (2) e.g.) ? More precisely consider a tempered distribution $u$ over $Omega$ and:
begin{equation}
|u|_{dot{H}_*^{-1}}:= left(int_{Omega} |omega|^{-2}|hat{u}(omega)|^{2} d omegaright)^{1/2}
end{equation}

where $hat{u}$ writes for the Fourier transform. My idea is that if $nu$ is a signed measure zero total mass and with density wrt the Lesbegue measure, then it can be written as $nu= (f-g) dx$ where $f,g$ are positive functions with $int f dx=int g dx$. It can be seen as a the following tempered distribution:
begin{equation}
forall phi in S(Omega), langle nu,phirangle=int_{Omega} phi(x)d nu(x)=int_{Omega} phi(x)(f(x)-g(x))dx
end{equation}

where $S(Omega)$ is the Schwartz class. Moreover we would have something like $hat{nu}(omega)= (hat{f}(omega)-hat{g}(omega))$.

In a dirty way we would have also $|nabla h|_2^{2}=|omega|^{2} |hat{h}(omega)|^{2}$ so $|h|_{dot{H}^{1}}=| |omega| hat{h} |_{L_2}$. Moreover $|langle h,f-grangle|=|langle hat{h},hat{f}-hat{g}rangle|$ by Plancherel. And also $|langle hat{h},hat{f}-hat{g}rangle|=|langle |omega|^{1} hat{h},|omega|^{-1}(hat{f}-hat{g})rangle|leq | |omega|^{2} hat{h} |_{L_2} | |omega|^{-2}|hat{f}-hat{g}|^{2}|_{L_2}$. This would give:
begin{equation}
|nu|_{dot{H}^{-1}}= left(int_{Omega} |omega|^{-2}|hat{f}(omega)-hat{g}(omega)|^{2} d omegaright)^{1/2}=|nu|_{dot{H}_*^{-1}}
end{equation}

Does this reasoning make sense ? Does it requires additional assumptions on $h,f,g$ ?

(1) Comparison between W2 distance and H˙ −1 norm, and localisation of Wasserstein distance. Rémi Peyre. 2018.

(2) Fourier Analysis and Nonlinear Partial Differential Equations, Hajer BahouriJean-Yves CheminRaphaël Danchin. 2011

fa.functional analysis – Strict Riesz’s rearrangement inequality

(Continue to the last question Riesz rearrangement inequality) In the Lieb-Loss’s book , they present the strict Riesz rearrangement inequality in Section3, Theorem 3.9(Page 93). They say that when the functions f,g,h are all nonnegative, and if g is strictly symmetric decreasing, then Riesz rearrangement inequality holds and the “=” holds iff f and h are translation of $f^*, h^*$. Namely if f,g,h are all nonnegative, then
$$iint_{mathbb{R}^ntimes mathbb{R}^n} f(x) g(x-y) h(y) , dx,dy \
le iint_{mathbb{R}^ntimes mathbb{R}^n} f^*(x) g^*(x-y) h^*(y) , dx,dytag{1}$$

and if g is strictly symmetric decreasing, then there is a equality only of $f=T(f^*), h=T(h^*)$ for some translation $T$. I want to ask when remove the nonnegative condition, such as g(x)=-ln(x), whether the “=” holds iff f and h are translation of $f^*, h^*$. For example, let $g(x)=-ln x$, which is strictly symmetric decreasing. In this cases, we know that (1) still holds. Does the equality holds in (1) only if f and h are a translation of $f^*, h^*$?

fa.functional analysis – Solution of SDE system with each equation having strong unique solution

Suppose that we have a system of SDEs

$$
d X_t = a(X_t, Y_t)dt + b(X_t, Y_t) dB^x_t,\
d Y_t = c(Y_t)dt + d(Y_t) dB^y_t,
$$

Also let

  1. The initial condition $X_0$, $Y_0$ and filtrations generated by $B^x_t$, $B^y_t$ are mutually independent. $B^x_t$ and $B^y_t$ are also independent.
  2. $Y_t$ has strong unique solution with respect to $Y_0$ and $B^y_t$.
  3. For every realization of $Y_t$ (i.e., fix $Y_t$), the solution of $X_t$ is strong and unique.

Question is, is the solution to the entire SDE system is also strong and unique? If not what other conditions are missing?

I think the underlying solution probability space is the Cartesian of the solution probability spaces of these two sub-SDEs, should it be?

fa.functional analysis – lifting theorem for n case

I am aware of the following statement of the lifting theorem. For i∈{1,2} let Bi be a contraction on a Hilbert space Hi and let Ai, acting on the Hilbert space Ki, be the minimal unitary dilation of Bi. Let Pi be the orthogonal projection of Ki onto Hi. Then an operator X from H1 to H2 satisfies B2X=XB1 if and only if there exist an operator Y from K1 to K2 such that

A2Y=YA1,
∥X∥=∥Y∥,
P2YP1=XP1.
I am looking for a similar theorem but for i∈{1,2,…n}. So now X satisfies n operator equations and I want a lift Y of X which will further satisfy n equations. Any reference /suggestion are most welcome for the above question.

fa.functional analysis – A convex function is “usually” subdifferentiable

Let $X$ be a locally convex topological vector space, and let $f:Xtomathbb Rcup{infty}$ be a proper, convex, lower semicontinuous function, whose effective domain $D:=f^{-1}(mathbb R)$ is compact. Let $Ssubseteq D$ be the set of points at which $f$ is subdifferentiable.

My question is somewhat vague: What results are there that say $S$ constitutes “most of” $D$? One example of such a result is the Brøndsted-Rockafellar theorem, which says $S$ is dense in $D$ whenever $X$ is a Banach space.

Ideally, I’d like to know any results guaranteeing $S$ is most of $D$ in some sense other than topological (e.g. measure-theoretic). Any pointers to such results would be very helpful.

fa.functional analysis – Is the bitranspose continuous for the $sigma$-strong topology?

Let $varphicolon Ato B$ be a bounded, linear map between C*-algebras. Is the bitranspose $varphi^{**}colon A^{**}to B^{**}$ continuous when the von Neumann algebras $A^{**}$ and $B^{**}$ are equipped with their $sigma$-strong topologies?

Motivation/Background: Note that $varphi^{**}$ is clearly continuous when $A^{**}$ and $B^{**}$ are equipped with their $sigma$-weak topologies, since these agree with the weak${}^*$-topology from the preduals, and $varphi^{**}$ is weak${}^*$-continuous (that is, $sigma(A^{**},A^*)-sigma(B^{**},B^*)$-continuous).

If $varphi$ if completely positive, then it follows that $varphi^{**}$ is a completely positive, normal map, and therefore is continuous for the $sigma$-strong topologies. Thus, the question is only interesting if $varphi$ is not completely positive.

On bounded sets of a von Neumann algebra, the $sigma$-strong topology agrees with the strong (operator) topology (SOT). If $Msubseteq B(H)$ is a von Neumann algebra, then a net $(a_j)_j$ in $M$ SOT-converges to $ain M$ if $|a_jxi-axi|to 0$ for every $xiin H$.

fa.functional analysis – Nonvanishing section of infinite-dimensional tautological bundle

Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a half-dimensional subspace as a subspace $W subset H$ such that both $W$ and $W^perp$ have infinite dimension.

Fix one half-dimensional subspace $W_0$. The Grassmannian of $H$ is
$$mathrm{Gr}(H, W_0) = {W subset H ~|~ W text{ is half-dimensional}, P_W – P_{W_0} text{ is Hilbert-Schmidt}}. $$
Here for $W subset H$ a subspace, $P_W$ denotes the orthogonal projection onto $W$. $mathrm{Gr}(H, W_0)$ can be given the structure of a Hilbert manifold in a natural way (see e.g. the book “Loop Groups” of Pressley and Segal).

The space $mathrm{Gr}(H, W_0)$ has a tautological vector bundle $tau$ over it, where the fiber is given by $tau(L) = L$.

Question: Does $tau$ have a nowhere vanishing section?

I believe that in the case that $H$ is finite-dimensional (say of dimension $2n$), the answer is no, as one can show that the Euler class of $tau$ is non-zero. But how would one proceed in the infinite-dimensional case?

fa.functional analysis – Where can I find general information about the RKHS of a Gaussian kernel?

Let $k$ be the Gaussian, or squared-exponential kernel on $mathbb R^n$:

$$k(x, y)=e^{-clVert x-yrVert_2^2}$$

where $c$ is some positive constant. This kernel has a reproducing kernel Hilbert space $H$ associated with it. I’m interested mainly in topological questions such as:

  • Is this space dense in other spaces, such as the space of continuous functions, or $L_2$?
  • How does the norm of $H$ interact with its topology? For example, does it explode at the boundary of $H$, i.e., if $f_nto f$ with $f$ not in $H$, does $lVert f_nrVert$ explode?

However, general results about $H$, or about RKHSs in general, which would be useful in attacking questions such as these, would also be useful.

fa.functional analysis – Eigenvalues of splitting scheme

In numerical analysis it is common to approximate a solution to a PDE

$$u'(t) = (A+B) u(t), quad u(0)=u_0$$

which is just given by $e^{(A+B)t}u_0$ by the splitting $e^{B/2 t} e^{At} e^{B/2t}u_0.$ Here, $A,B$ can be assumed to be self-adjoint matrices.

We shall furthermore assume that $A+B$ generates a contraction semigroup, i.e. $Vert e^{(A+B)t} Vert le 1,$i.e. all eigenvalues of $A+B$ are negative.

Since structure preserving schemes are important, I am looking for a criterion such that $$Vert e^{B/2 t} e^{At} e^{B/2t}Vert le C text{ for all }t in (0,infty).$$

There are obviously some conditions like $A,B$ individually having only non-positive eigenvalues etc. that would do the job, but I am looking for a more intelligent criterion that takes into account the fact that this is really supposed to be an approximation of the exponential of $A+B$.

fa.functional analysis – How to characterize the order convergence in Bochner-integrable functions space?

Let $(Omega,Sigma,mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(mu,X)$, $X$ Banach lattice.

In $L^p$ we have: A sequence $(f_n)_1^inftysubset L^P(mu)$ is order convergent to $f$ as $ntoinfty$ if and only if there exists some $0leq gin L^p(mu)$ such that $|f_n|leq g$ a.e. and $f_nto f$ a.e.

We want a similar result for $L^1(mu,X)$: If $(f_n)_1^inftysubset L^1(mu,X)$ is a sequence of functions, with $X$ $sigma$-order continuous. Then $(f_n)$ is order convergent to $f$ as $ntoinfty$ if and only if there exists some $0leq gin L^1(mu,X)$ such that $|f_n|leq g$ a.e. and $f_nto f$ a.e.

This is my try:

$Rightarrow)$ Suppose $f_nxrightarrow{o} f$. Then, exists $g_ndownarrow 0$ such that $|f_n-f|leq g_n$ for all $ninmathbb{N}$. Since $X$ is $sigma$-order continuous, $g_ndownarrow 0$ implies $|g_n|downarrow 0$ a.e., then $|g_n|to 0$ a.e. Hence $g_nto 0$ a.e. Thus $f_nto f$ a.e. On the other hand, note that $|f_n|=|f+f_n-f|leq |f|+|f_n-f|leq |f|+g_nleq |f|+g_1$. Define $g=|f|+g_1geq 0$ we done.

$Leftarrow)$ Suppose, without loss of generality, $f_nto 0$ a.e. and there exists $gin L(mu,X)$ such that $|f_n|leq g$. Define $g_n=sup{|f_m|:mgeq n}$ for each $ninmathbb{N}$. Note that $g_ndownarrow$. Since $X$ is $sigma$-order continuous, $L^1(mu,X)$ is $sigma$-Dedekind complete, so $g_nin L^1(mu,X)$ for every $ninmathbb{N}$.

If $f_nto 0$ a.e., then $g_nto 0$ a.e.? Is this true? (*)

Since $g_nleq g$ and $g_nto 0$ a.e., by the monotone convergence theorem, $g_nto 0$. Thus $f_nxrightarrow{o}0$.

Some notes and comments:

  • First, (*) is true? I have not been able to prove it.
  • My proof is wrong? Some alternative or new ideas?
  • If $X$ is $sigma$-order continuous, then $L^1(mu,X)$ is $sigma$-Dedekind complete. This is a proven fact.

Notation, definitions, etc:

  • $L^1(mu,X)={f:Omegato X , |, f text{ is Bochner integrable}}.$
  • $g_ndownarrow 0$ means $g_nleq g_{n+1}$ for all $ninmathbb{N}$ and $inf_n g_n=0$.
  • $X$ is $sigma$-order continuous if $x_ndownarrow 0$ implies $|x_n|downarrow 0$.
  • $X$ is $sigma$-Dedekind complete if $(x_n)subset X$ is a bounded sequence implies $sup_nx_n,inf_nx_nin X$.
  • $X$ Banach lattice, for all $xin X$ we define $|x|=sup{-x,x}$,.
  • In a Riesz space, a sequence $(x_n)subset X$ is called order convergent to $x$ if there exists a sequence $(y_n)subset X$ such that $y_ndownarrow 0$ and $|x_n-x|leq y_n$ for all $ninmathbb{N}$. We write $x_nxrightarrow{o}x$.

Thanks in advance.