I am trying to show the following two statements are true:

(1) For any nonempty set $\Omega\subset\mathbb{R}^n$, the set $B$ consisting of points $y\notin\Omega$ where there is not a unique closest point $x\in\partial\Omega$ for each $y$ has $\mathcal{H}^n$-measure 0.

(2) This implies that the number of points $a\in A\subset \mathbb{R}^2$ such that $\mathcal{H}^{1}(\vec{n}(a)\cap B)>0$ is countable, where $\vec{n}(a)$ is the normal to $a\in A$ and $A$ is $C^1$ (i.e. it is locally the image of a $C^1$ function from $\mathbb{R}$ to $\mathbb{R}$).

For (1), I would like help proving only this related fact: if the distance function $f$ to $\Omega$ is differentiable at a point $y\notin\Omega$, then there exists a unique closest point $x\in\partial\Omega$ to $y$.

I thought that I could get at this by saying that the graph of $f$ has a sharp corner wherever there is more than one closest point (and hence is not differentiable there). But, I am told this is not always true.

For (2), I thought that, since we're in $\mathbb{R}^2$, if we have an uncountable number of segments with positive $\mathcal{H}^{1}$ measure, then $B$ would have to have positive $\mathcal{H}^{2}$ measure since $B$ contains these segments. I'm told this is not true either. Please assist.

I have a question about semigroup of one-dimensional diffusions.

Let $X$ be the Ornstein Uhlenbeck process on $\mathbb{R}$. It is known that the $L^2$ semigroup $\{T_t\}$ of $X$ is a compact operator on $L^{2}(\mathbb{R},m)$, where $m$ is the speed measure of $X$. $\{T_t\}$ is extended to strongly continuous contraction semigroup on $L^{p}(\mathbb{R},m)$, $1\le p<\infty$. Moreover, $\{T_t\}$ becomes a compact operator on $L^{p}(\mathbb{R},m)$ for any $1<p<\infty$. $\{T_t\}$ is not a compact operator on $L^{1}(\mathbb{R},m)$.

**My question**

- Is there are nontrivial diffusion on $\mathbb{R}$ whose semigroup is compact on $L^{p}(\mathbb{R})$ for any $1 \le p \le \infty$? Consider the following differential operator: \begin{equation*} \frac{d^2}{dx^2}-x^{3}\frac{d}{dx}. \end{equation*} and the diffusion $Y$ associated with the above operator. The semigroups associated with $Y$ is compact on $L^{p}(\mathbb{R},m)$ for any $1 \le p \le \infty$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary.

- What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$ Can one find a solution in $W^{1,1}_0(\Omega)$? (I think the answer is no, but I do not know a counterexample). I know that $-\Delta u=f\in L^1$ has a unique solution in $W^{1,p}_0(\Omega)$ for all $1\leq p<n/(n-1)$, (see https://mathoverflow.net/a/298962/121665) but now the right hand side is much worse.
- What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^p(\Omega,\mathbb{R}^n), 1<p<\infty? $$ If $1<p<\infty$ we can solve the quation $-\Delta U=F$, $U\in W^{2,p}(\Omega)$. Then $u={\rm div}\, U\in W^{1,p}$ is a solution to $-\Delta u={\rm div}\, F$, but I do not know if we can take $u\in W^{1,p}_0(\Omega)$.

My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions

Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,1)$ is the unit ball of $\mathbb{R}^4$ and for a fixed $n\geq 1$. We define the bi-energy $B$ as: $$B(u):=\int_{B(0,1)} \vert \Delta u\vert^2 \, dx $$

(or $B(u):=\int_{B(0,1)} \Vert (\Delta u)^T \Vert^2 \, dx $ , where $(\Delta u)^T$ is the projection of $\Delta u$ on $T_u S^n$).

In fact it is quite important to notice that

$$\Vert (\Delta u)^T \Vert^2= \Vert \Delta u\Vert^2 -\Vert \nabla u\Vert^4 \; (1)$$

Critical points of this equation are known as biharmonic maps, generalizing harmonic in dimension $4$. The Euler-Lagrange equation is quite complicated see the introduction of [1], but it is mostly like $$\Delta^2 u= \langle \nabla u, \nabla \Delta u\rangle + \vert \nabla^2 u\vert^2 + \vert\nabla u\vert^2 \Delta u +\vert \nabla u\vert^4 $$

In the article they prove an $\epsilon$-regularity result, that is to say if the $W^{2,2}$-norm of $u$ is small enough (in fact in the sphere case $B(u)$ is enough because of the remark above) then you control all the derivatives on $B(0,1/2)$. It is a very powerful result. But I wonder about a global one. Of course you can't expect to control anything more than $W^{2,2}$ on $B(0,1)$ but at least can you prove that $\Delta^2u$ is in $L^1$?

Here is the precise question, does there exist $\epsilon >0$ such that if $\phi\in W^{2,2}$ with $B(\phi)\leq \epsilon$ and $u\in W^{2,2}(B(0,1),S^n)$ is a minimizer of $B$ under the boundary condition $u=\phi,\partial_\nu u=\partial_\nu \phi$ on $\partial B(0,1)$, then $$\Delta^2 u \in L^1$$ or even $$\Vert \Delta u\Vert_1\leq f(\epsilon)$$ with some $f\in C(\mathbb{R}^+)$ with $f(0)=0$.

It would be very surprising to me if not since it is for free for harmonic maps since the equation is $\Delta u =\vert \nabla u\vert^2$, hence the $L^1$-norm of the Laplacian is automatically controlled by the energy.

Informally the idea of this question is about whether the rules of set theory can be derived as a transfer of some rules from the hereditarily finite set realm, and whether this transfer principle itself can be coined for notions other than the "finite" notion?

The principle I want to negotiate is: "if $\phi$ is a property that is definable by a formula in the language of set theory that is strictly shorter than the shortest parameter free formula in that language that can define 'finiteness', then if $\phi$ is CLOSED on the the hereditarily finite set world, then it can be generalized over the whole realm of sets"!

The crude informal idea is that if a property that cannot mention finiteness generalizes over the whole hereditarily finite set realm, then it can go beyond it.

To formally capture that, I'll work up in a class theory, so we define "set" as an element of a class, the language of the theory is mono-sorted first order logic with identity and membership, we stipulate axioms of:

- Extensionality: as in ZF
- Class comprehension schema: $\forall x_1,..,x_n\ \exists x \ [x=\{y|set(y) \wedge \phi(y,x_1,..,x_n)\}] $
- The empty class is a set
- Singletons: $\forall x [ set(x) \to set(\{x\})]$
- Boolean Union: $\forall x,y [set(x) \wedge set(y) \to set (x \cup y)]$

Define: $$ fin(A) \iff \forall K (\exists o (o \in K \wedge \neg \exists m (m \in o)) \wedge \forall x (x \in A \to \exists y (y \in K \wedge x \in y \wedge \forall z (z \in y \to z=x))) \wedge \\ \forall a \forall b (a \in K \wedge b \in K \to \exists c (c \in K \wedge \forall d (d \in c\leftrightarrow d \in a \lor d \in b ))) \\ \to A \in K)$$

In English: $A$ is finite if and only if it is an element of every class $K$ that contains the empty set among its elements, is closed under Boolean union and that has the singletons of all elements of $A$ among its elements.

"I think this is along the shortest way to define "finite set" in the first order language of set theory"

Perhaps the above formula can be shortened further, or perhaps there is another shorter parameter free formulation of "x is a finite set" in the language of set theory, however for the sake of presentation here we'll take this formula to be the shortest formula defining finiteness.

- $ \forall x (x \text{ is hereditarily finite } \to set(x))$

Where "x is hereditarily finite" is defined as the transitive closure class of x being finite.

We shall denote the class of all sets by $V$, and the class of all hereditarily fintie sets by $HF$

$HF \in V$

**The principle of Transfer from the pure finite world:**if $\phi(y,x)$ is a formula shorter than any formula defining finiteness, in which only symbols $``y,x"$ occur free, and those only occur free, then:

$\forall x [x \in HF \to \forall y (\phi(y,x) \to y \in HF)] \to \forall x \in V [ \forall y (\phi(y,x) \to y \in V)]$

Now it is clear that all axioms of Union, Power and Separation over sets are derivable from the above transfer principle, and so $ZC$ is interpretable here. Actaully if we restrict $\phi$ to have no more than three atomic subformulas, we can still interpret the whole of $ZC$. Replacement is not interpretable by this principle. Yet, a minor modification of this principle might succeed in proving replacement over sets, this can be done by changing the closure property to involve only subsets of $HF$, what I call as "proximity closure over HF", so to restate that:

8'.**The principle of Transfer from proximity of the pure finite world:** if $\phi(y,x)$ is a formula shorter than any formula defining finiteness, in which only symbols $``y,x"$ occur free, and those only occur free, then:

$\forall x [x \in HF \to \forall y \subseteq HF (\phi(y,x) \to y \in HF)] \to \forall x \in V [ \forall y (\phi(y,x) \to y \in V)]$

That replacement is provable can be shown from examining the following formula whose length is shorter than any formula defining finiteness.

$\exists F [\forall m (m \in F \to \exists a,b (a \in A \wedge b \in B \wedge a \in m \wedge b \in m)) \wedge \\ \forall m,n (m \in F \wedge n \in F \wedge \exists k(k \in m \wedge k \in n) \to n=m) \wedge \\ \forall b (b \in B \to \exists m (m \in F \wedge b \in m))]$

Now if $A$ is hereditarily finite and $B$ is a subset of $HF$ that fulfills the above formula, then $B$ is hereditarily finite, this mean that the property defined by the above formula is "proximity closed over the hereditarily finite world"

I don't have any proof of consistency of these principles, but if there is no clear inconsistency of those relative to $ZF$ or $MK$ or some extension of those, then could it be possible to think of extending that principle to properties other than "x is finite"? so we generalize it to some line properties, so for a property $P$ in that line, we stipulate that:

*any predicate $Q$ that is closed over the pure $P$ world, would generalize over the whole set world,*

or even stronger:

*any predicate $Q$ that is proximity closed over the pure $P$ world, would generalize over the whole set world*

Of course in both cases $Q$ must be expressible by a formula strictly shorter than the shortest expression defining property $P$, and also we stipulate parallel axioms sufficient to define the property $P$, also axioms asserting the element-hood of all hereditarily $P$ classes, and the existence of a set of all hereditarily $P$ sets. Of course this can only be done for some selected line of properties.

is that possible or it is involved with clear inconsistencies? and what would be the general qualification of such property $P$?

Are there any examples where a countable support iteration of proper forcing defined in an inner model does crazy things in an outer model? This is vague, so to narrow it down, are there examples of $V\supset M$ and a countable support iteration of proper forcing $\langle P_i, \dot{Q}_j: i\leq \omega, j<\omega\rangle$ defined in $M$ such that for each $j\in \omega$, $P_j$ is proper in $V$, but $P_\omega$ collapses $\omega_1$ in $V$. Of course, when $V$ and $M$ are sufficiently close (for example $V\models M^\kappa\subset M$ for large enough $\kappa$), the iteration will look the same. Since we are taking countable support iteration, I'll impose that (at least) $V\models M^\omega\subset M$.

Another test question:

Suppose $V\models M^\omega \subset M$. Is it true that countable support iteration of Cohen forcing defined in $M$ is equivalent to the one defined in $V$?

Note: this question was previously asked at math.stackexchange (under the same title) to no avail.

I have been recently reading Dr. Kaledin's notes on algebraic geometry. There is a statement in lecture 16 about which I feel confused.

Оказывается, что для пучков на проективном пространстве, полином Гильберта это единственный существенно дискретный инвариант: как только он зафиксирован, можно построить такое плоское семейство над конечномерной нётеровой базой $Y$, что любой пучок с данным $P(\mathcal{F},l)$ появляется в нем как слой, причем только один раз.

It can be translated as "It turns out that for sheaves on projective space Hilbert polynomial is the only essentially discrete invariant: once it is fixed, it is possible to construct a flat family over a finite-dimensional Noetherian base Y such that any sheaf with given $P(\mathcal{F}, l)$ appears in it as fiber exactly once".

I struggle to translate this remark into a precise mathematical statement. In particular, I don't understand what does 'exactly once' mean; if we have two points $y_1, y_2 \in Y$, how can we compare sheaves on $f^{-1}(y_1)$ and $f^{-1}(y_2)$? Can someone provide me the precise statement that Kaledin probably had in mind?

I am looking for a graph-theoretic algorithm, that determines among all simple paths $\mathcal{P}_{ab}$ that connect vertex $a$ with vertex $b$ the one, that has minimal average edge-length, i.e. $$\ p_{ab}\in\mathcal{P}_{ab}: \frac{\ell(p_{ab})}{card(p_{ab})}\ \le\ \frac{\ell(q_{ab})}{card(q_{ab})}\ \forall q_{ab}\in\mathcal{P}_{ab}$$

By "graph-theoretic" I mean algorithms in the vein of those that are likely to be found in publications about algorithmic graph-theory; on the contrary, mathematical programming is not what I am looking for.

First consider Schubert cells in the full flag variety $Fl_n$. Then for a permutation $w\in S_n$, the poset of Schubert varieties contained in the Schubert variety corresponding to $w$ is isomorphic to the interval $[e,w]$ in the *strong* Bruhat order.

Now consider Schubert cells in the Grassmannian $Gr(k,n)$. Then for a Grassmannian permutation $w\in S_n$, the poset of Schubert varieties contained in the Schubert variety corresponding to $w$ is isomorphic to the *dual* $[e,w]^*$ of the interval $[e,w]$ in the *weak* Bruhat order.

Is there some intuitive explanation for this duality between strong and weak Bruhat orders?

Let $E$ be a metrizable locally convex topological vector space and let $E^{*}$ be its dual space endowed with the strong topology = topology of uniform convergence on (closed convex balanced) bounded subsets of $E$. Let $F\subset E^{*}$.

Is it true that $\overline{F}=F_1$, where $F_1$ is the set of all linear functionals on $E$, whose restrictions on every (closed convex balanced) bounded subset of $E$ are continuous in the weak $\sigma(E,F)$ topology?

It seems that I can adapt the proof of Grothendieck's completion theorem to prove this. Indeed, every linear functional continuous on a closed set $B$ in $E$ can be uniformly approximated on $B$ by an element of $E^{*}$ (I guess this is Grothendieck's lemma). Hence, we only need to show that $F_1\subset E^{*}$. But this follows from the fact, that since $E$ is metrizable, $E^{*}$ is complete, and so it is a closed subset of $E'$ (the algebraic dual), andowed with the uniformity of uniform convergence on bounded sets.

If this is correct, I hope this result is contained in some textbook on locally convex spaces, and so a reference would be highly appreciated.

I stumbled upon this while I was working on something related to the Wasserstein metric and Gaussian distributions. Maybe it actually very easy, but I could not find how to look it up.

On $\mathbb{R}^n$, take any convex function $\phi$, and denote its distributional gradient with $\nabla \phi$. Consider measure $\mu = \nabla\phi_{\#} N(0,I)$, where $N(0,I)$ is the distribution of a normal Gaussian random vector. Let $\Sigma_{ij}$ denote the square root of the covariance matrix of $\mu$.

I am wondering whether the following is true: $$ \Sigma_{ij} = \int_{\mathbb{R}^n} x_i (\nabla\phi(x))_j dN(0,I)(x) $$

In the 1-d case, this means: for any nondecreasing f, we should have $$ \int f(x)^2 dN(0,1)(x) - \left(\int f(x) dN(0,1)(x)\right)^2 = \left(\int x f(x) dN(0,1)(x)\right)^2 . $$ Is it correct? If not, is the right-hand-side still somehow related to the variance?

Let $N$ be a positive inter and $p$ a prime not dividing $N$. Let $X_0(N)$ be the modular curve (over $\mathbb{Q}$) associated to the congruence subgroup $\Gamma_0(N)$, which is the subgroup of $\text{SL}_2(\mathbb{Z})$ consisting of upper triangular matrices modulo $N$. Let $\pi_1$ and $\pi_p$ be two degeneracy maps from $X_0(Np)$ to $X_0(N)$ defined by "forgetting the level $p$ structure" and "dividing by the level $p$ structure" on the associated moduli problem. If we identify $X_0(Np)(\mathbb{C})\simeq \Gamma_0(Np)\backslash(\mathbb{H} \cup \mathbb{P}^1(\mathbb{Q}))$ and $X_0(N)(\mathbb{C})\simeq \Gamma_0(N)\backslash(\mathbb{H} \cup \mathbb{P}^1(\mathbb{Q}))$, then we have

$$ \pi_1(z)=z \pmod {\Gamma_0(N)} ~~\text{ and }~~ \pi_p(z)=pz \pmod {\Gamma_0(N)}. $$ Let $w_p$ be the Atkin-Lehner involution on $X_0(Np)$ which satisfies $$ \pi_1\circ w_p = \pi_p. $$

Now, we want to understand how these maps send cusps explicitly. Let $C(Np)$ and $C(N)$ denote the sets of cusps on $X_0(Np)$ and $X_0(N)$, respectively. Then the maps $\pi_1$ and $\pi_p$ send $C(Np)$ to $C(N)$, and $w_p$ becomes an involution on $C(Np)$ by restriction. Let $$ S_1=\{ (\begin{smallmatrix} x \\ d \end{smallmatrix}) : d \mid N, ~(x, d)=1,~1\leq x \leq d, ~x \text{ taken modulo } (d, N/d) \} $$ and $$ S_2=\{ (\begin{smallmatrix} x \\ dp \end{smallmatrix}) : d \mid N, ~(x, dp)=1,~1\leq x \leq dp, ~x \text{ taken modulo } (dp, N/d) \}. $$ (Note that $(dp, N/d)=(d, N/d)$.) Then, we have $$ C(Np)=S_1 \cup S_2 ~\text{ and }~ C(N)=S_1 $$ (cf. Section 3.8 of Diamond-Shurman's book, A first course in modular forms). I think that from the description of the maps $\pi_1$ and $\pi_p$ on $X_0(Np)(\mathbb{C})$, we have $$ \pi_1((\begin{smallmatrix} x \\ d \end{smallmatrix}))=(\begin{smallmatrix} x \\ d \end{smallmatrix}) \text{ and } \pi_p((\begin{smallmatrix} x \\ d \end{smallmatrix}))= (\begin{smallmatrix} y \\ d \end{smallmatrix}), $$ where $y$ is the remainder of $px$ divided by $d$. Also, we have $$ \pi_1((\begin{smallmatrix} x \\ dp \end{smallmatrix}))=(\begin{smallmatrix} z \\ d \end{smallmatrix}) \text{ and } \pi_p((\begin{smallmatrix} x \\ dp \end{smallmatrix}))= (\begin{smallmatrix} z \\ d \end{smallmatrix}), $$ where $z$ is the remainder of $x$ divided by $d$. I guess that $w_p$ gives rise to a bijection between $S_1$ and $S_2$.

I have two questions:

Q1. Are the above formulas correct? If not, what are the correct ones?

Q2. What is the explicit description of $w_p$? In other words, what are $$ w_p((\begin{smallmatrix} x \\ d \end{smallmatrix})) ~\text{ and }~ w_p((\begin{smallmatrix} x \\ dp \end{smallmatrix}))? $$

So far, I have tried to compute them using $\pi_1 \circ w_p=\pi_p$, but I failed. I hope someone can help me to understand this problem.

In a DG category $\mathcal{C}$, call a (closed degree 0) morphism $f: X \to Y$ a homotopy equivalence, if there exists $g: Y \to X$ such that $gf-1_X$ and $fg-1_Y$ are exact.

For a full DG subcategory $\mathcal{D} \subset \mathcal{C}$, recall the explicit construction of Drinfeld quotient $\mathcal{C}/\mathcal{D}$: the objects are that of $\mathcal{C}$ and the morphisms are freely generated by the morphisms of $\mathcal{C}$ and, for every $X \in \mathcal{D}$, by $b_X: X \to X$ of degree $-1$ with $d(b_X)=1_X$.

Now consider the following procedure of "homotopically inverting" a chosen (closed degree 0) morphism $f$ in $\mathcal{C}$. Take the Joneda embedding $J: \mathcal{C} \to \mathcal{C}\mathrm{Mod} = \mathrm{DGFun}(\mathcal{C}^{\mathrm{Op}},Ch_k)$; in $\mathcal{C}\mathrm{Mod}$ there exists $\mathrm{Cone}(Jf)$, so pass to $\mathcal{C}\mathrm{Mod}/\mathrm{Cone}(Jf)$ and consider there a full subcategory on the objects of $J\mathcal{C}$. Explicitly, this means freely adding to $\mathcal{C}$ the maps $f':Y \to X$ of degree 0 with $d(f')=0$, $r_X: X \to X$ of degree $-1$ with $d(r_X)=f'f-1_X$, $r_Y: Y \to Y$ of degree $-1$ with $d(r_Y)=ff'-1_Y$ and $r_{XY}: X \to Y$ of degree $-2$ with $d(r_{XY})=fr_X-r_Yf$. Denote the result by $L_H(\mathcal{C},f)$. (For example, applying this to the single arrow in the category $\bullet \to \bullet$ gains a category quasiequivalent to a point which came up in Tabuada's construction of the model structure for DG categories.)

Now, for an arbitrary DG category $\mathcal{A}$, compare the following DG categories:

- a full subcategory in $\mathrm{DGFun}(\mathcal{C},\mathcal{A})$ consisting of DG functors that take $f$ into a homotopy equivalence in $\mathcal{A}$;
- $\mathrm{DGFun}(L_H(\mathcal{C},f),\mathcal{A})$.

These two categories are not the same but my expectation is that they are queasiequivalent.

My question is: are they?

Given a D-module $\mathcal{M}$ on $\mathbb{C}^n$, its Fourier transform $\widehat{\mathcal{M}}$ is equal to $\mathcal{M}$ as a set, but its module structure over $\mathbb{C}[x_1,...,x_n,\partial_1,...,\partial_n]$ is given by $$x_i \cdot m \ = \ -\partial_i m \ \ \ \text{ and } \ \ \ \partial_i \cdot m \ = \ x_im$$ This definition comes from how the classical Fourier transform acts on ODEs, see *.

But now consider a compact Lie group $G$. For each $\rho \in \widehat{G}$ there is a ``Fourier transform'' $$\text{functions on }G \ \longrightarrow \ \mathbb{C}$$ $$f \ \longrightarrow \ \int f(g)\rho(g) dg$$

Is there a ``Fourier transform'' of D-modules on $G$ (or maybe some flag variety of $G$) associated to each $\rho \in \widehat{G}$?

The reason I suspect this might be true is that, though one might initially protest that this is probably an affine phenonemon, Lie groups (or rather their flags) are D-affine.

$$\text{}$$ $$\text{}$$

*On $\mathbb{C}$, the Fourier transform take an ODEs with polynomial coefficients $$a_n(x)\partial^nf+\cdots+a_0(x)f \ = \ 0$$ to another such ODE $$a_n(-\partial)k^n\widehat{f}+\cdots+a_0(-\partial)\widehat{f}\ = \ 0$$ because the Fourier transform of $x^m$ is $\delta^{(m)}(k)$. The same is true for $\mathbb{C}^n$.

Let's say we know that a random variable $X$ satisfies
$$
E\big[ e^{a(\log |X|)^2} \big] < \infty
$$
for $a \in (0,c)$, where $c$ is some positive constant. This is clearly weaker than having *exponential moments* (i.e.
$
E[e^{aX}]<\infty
$). Is there a name we give this property?

The Hardy-Littlewood maximal function of a function $f$ is defined by $$ M f(x):=\sup_{0<r<\infty}\frac{1}{|B_r|}\int_{B_r}|f(x+y)|dy, $$ where $|B_r|$ denotes the Lebesgue measure of the ball $B_r$. It is well-known that for $p\in(1,\infty]$, there exists a constant $C_{d,p}>0$ such that \begin{align} \|M f\|_p\leq C_{d,p}\|f\|_p. \label{mf} \end{align} That is, $M$ is bounded in $L^p$-spaces.

The zero order Besov space can be defined as follows. Recall the Littlewood-Paley operators is deifned as $$ \Lambda_jf:=h_j*f \quad (convolution), $$ where $h_j\in C_0^\infty(R^n)$. We have $$ f=\sum_{j\geq -1}\Lambda_jf. $$ Then, $B^0_{p,\infty}(R^n)$ is defined as $f$ satisfying $$ \sup_{j}\|\Lambda_jf\|_p<\infty. $$

Is the operator $M$ bounded in $B^0_{p,\infty}(R^n)$? (the problem I met is the change of order for $M$ and convolution. Moreprecisely, is it true $$ \Lambda_jMf\leq M\Lambda_jf?? $$ If this is true, then $\|\Lambda_jMf\|_p=\|M\lambda_jf\|_p\leq\|\lambda_jf\|_p$, this means that $\|Mf\|_{B^0_{p,\infty}}\leq C_d\|f\|_{B^0_{p,\infty}}$.)

In [1] (page 2) the authors say that finding the minimum hitting set / minimum set cover, in the case where all elements of the set belong to at most two of the specified subsets (the 2-hitting set problem), is equivalent to finding a maximal matching between the elements of the set (vertexes) and the set we need to hit (hyper-edges). I cant see how one problem can be mapped to the other - can anyone explain this?

Background: The minimum set cover / minimal hitting set problem can be visualised as a graph with vertices given by the set elements $\{v_1, v_2, \dots, v_n\}$ and hyper-edges given by the subsets that we need to hit, e.g. if there is a subset $C = \{v_2, v_3 \}$ that must be hit by the minimum hitting set, then $v_2$ and $v_3$ are joined by a hyper-edge.

[1] https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5533149

I'm trying to understand the Lemma 4.6 of Bhatt-Morrow-Scholze's paper **Integral $p$-adic Hodge Theory**.
In the proof, for proving the restriction functor is fully faithful, it used a affine open cover ${U_1,U_2}$ of the punctured spectrum $U$. But why it is enough to prove $A_{\inf} \cong \mathcal{O}(U)$ and $R=R_1\cap R_2\subset R_{12}$?

I have a sense that it is proving what we want on the affine subset, and then their intersection, but the detail is still a mess for me.

Also, why all vector bundle over $A_{\inf}$ is free?

I would be thankful if someone had references to provide...

I am a young researcher and sometimes I face an uncomfortable situation : I find an error in a research paper! Of course, most of the time, it is all just my misunderstanding but it happens that after checking tens of times, there is indeed an error and therefore I am not sure what I should do. "Am I the first one that notices this mistake or someone has already reported it? and then where?" "Hass this error been corrected in a later version? How can I know? ", "Should I send an email to the authors ? But still how could I be diplomatic enough not to bother them (and maybe the mistake is still mine, I am not an expert of the field)? " The situation gets worse when

-The paper is already old. Personally I would find annoying if someone asked me questions on a project finished years ago. And for very old papers the authors can just be retired already.

-It comes from a different field like physics or chemistry. Where they use a different language. How could I explain myself correctly and without creating a "diplomatic crisis"?

So I am here to ask advice to the experimented researchers how to deal with this awkward situation.