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Exercise generalizing (related to) Hölder's inequality

I came across this exercise and feel absolutely stuck: Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
HZA's user avatar
  • 1
2 votes
0 answers
104 views

Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support

This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction? Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
H A Helfgott's user avatar
  • 20.2k
0 votes
0 answers
64 views

$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable

Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported. I am wondering if we can ...
Jens Fischer's user avatar
4 votes
1 answer
273 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
90 views

About BMO space on smooth open bounded domain

Let $\Omega$ be any open domain in $\Bbb R^d$. Define the $\text{BMO}(\Omega)$ space as $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
Guy Fsone's user avatar
  • 1,101
3 votes
0 answers
82 views
+50

Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps

Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function $$ m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)}, $$ where $\lambda_{\max}$ denotes the largest eigenvalue....
Ran's user avatar
  • 73
2 votes
0 answers
77 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
224 views

A deceptively simple regularity problem for functions on the plane

By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer: Consider a twice ...
vmist's user avatar
  • 989
12 votes
1 answer
394 views

Is $X\times X$ homeomorphic to $X$ for a space of probability measures?

Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if $$ \int_S ...
Christian Remling's user avatar
3 votes
0 answers
67 views

Effective action of unbounded operators on subspaces outside their domains of definition

Consider a densely defined, self-adjoint operator $$ H: \mathcal{D} \rightarrow \mathscr{H}. $$ Assume for simplicity that $H$ is nonnegative. We want to effectively restrict this operator $H$ to a ...
Qualearn's user avatar
  • 133
0 votes
1 answer
157 views

Weak convergence of $f(x,e^{itx})$

This is the desired result (what I want to prove): $$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$ Given that $f\in C([a,b]\times\{e^{i\...
Quý Nhân's user avatar
9 votes
1 answer
292 views

What are the points of the algebra of polynomial functions on an arbitrary vector space?

Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
Dima Roytenberg's user avatar
0 votes
0 answers
46 views

Amenability of locally convex algebras

Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.). Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
Onur Oktay's user avatar
  • 2,605
1 vote
0 answers
85 views

Density of a subset of Schwartz space in the fractional Sobolev space

It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
Nirjan Biswas's user avatar
1 vote
1 answer
151 views

Is smoothness preserved under an isometric isomorphism?

Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
Tuh's user avatar
  • 113
2 votes
0 answers
62 views

Bessel spaces and Triebel Lizorkin

It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
Guillermo García Sáez's user avatar
1 vote
0 answers
58 views

duality of sobolev spaces. Representation of elements in the dual

I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
Alucard-o Ming's user avatar
0 votes
0 answers
42 views

questions on stochastic kernels and pushforward operator

Let $f:X \rightarrow \Delta (Y)$ and $g:X \rightarrow \Delta (X)$ be two kernels. For any bounded measurable function $h_Y:Y \rightarrow \mathbb{R},$ define $F(h_Y):X \rightarrow \mathbb{R}$ such that ...
andy's user avatar
  • 1
4 votes
0 answers
90 views

Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 835
0 votes
0 answers
90 views

How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?

I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak* -Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
Tuh's user avatar
  • 113
0 votes
1 answer
124 views

Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
Ali's user avatar
  • 4,135
1 vote
0 answers
146 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar
2 votes
0 answers
63 views

Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$

In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
Józef Zápařka's user avatar
0 votes
1 answer
239 views

Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]

Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
H Tomasz Grzybowski's user avatar
4 votes
0 answers
101 views

There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions

Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$. Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
Pierre PC's user avatar
  • 3,669
1 vote
0 answers
48 views

What makes the generalized projection different than metric on a Banach space?

I have came across the notion of generalized projection in Banach spaces, introduced by Ya. Alber and has seen many iterative algorithms being solved by using this projection. It helps in finding the ...
PPB's user avatar
  • 85
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
Liding Yao's user avatar
0 votes
0 answers
77 views

Nice formula for powers of modified Bessel function

Let $K_\nu(z)$ be the modified Bessel function of second kind. I am looking the geometric series $$1+aK_v+(aK_v)^2+(aK_v)^3...$$ I know there are formula for product of two such functions. I would ...
CO2's user avatar
  • 275
4 votes
0 answers
121 views

Projection behavior under the automorphism $\Phi$ on a diffuse semi-finite von Neumann algebra

Let $\mathcal{M}$ be a diffuse semi-finite von Neumann algebra acting on a Hilbert space $\mathcal{H}$, equipped with a faithful normal semi-finite trace $\varphi$. Let $\Phi : \mathcal{M} \to \...
DenOfZero's user avatar
  • 113
3 votes
0 answers
158 views

Gowers' dichotomy for quotients

Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable. A ...
M.González's user avatar
  • 4,461
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
52 views

On distributions and kernels

Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
G. Blaickner's user avatar
  • 1,429
0 votes
1 answer
169 views

Is the evolution family self-adjoint?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey ...
Akira's user avatar
  • 835
1 vote
1 answer
59 views

Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains

On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$. I'm interested ...
user2103480's user avatar
1 vote
0 answers
84 views

Does sets of positive capacity rule out constant functions?

Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by \begin{align*} \text{Cap}_{p}(K, U) := \inf \left\{ \int_U |\...
Guy Fsone's user avatar
  • 1,101
3 votes
1 answer
128 views

Comparing two different principles of premeasure-to-measure extension

It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for ...
Atom's user avatar
  • 133
2 votes
1 answer
236 views

Self-adjointness of generator and semigroup of an SDE

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 835
0 votes
0 answers
55 views

Compactness and Leray-Schauder degree

What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
Davidi Cone's user avatar
5 votes
1 answer
206 views

Compactness in trace class operators space

Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$. Are there easy ...
lulli_'s user avatar
  • 59
0 votes
0 answers
113 views

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...
S-F's user avatar
  • 63
2 votes
0 answers
70 views

Is the hypothesis "uniformly equivalent" needed?

I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows: Assume $\mathscr{H}$ is a Hilbert space of ...
ash's user avatar
  • 151
3 votes
1 answer
189 views

Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces

An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such ...
Xiangbo's user avatar
  • 33
0 votes
1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
charlie_beck's user avatar
3 votes
0 answers
95 views

Commutator of $A\otimes I$ and $I \otimes B$ vanishes?

Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
Hugo's user avatar
  • 31
6 votes
0 answers
212 views

Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck

In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
Cody Morrin's user avatar
1 vote
0 answers
111 views

References on the partial trace

For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows : $$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
Fawen90's user avatar
  • 1,389
0 votes
1 answer
188 views

Does the second Bourgain–Delbaen space belong to C_p?

The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$. An operator $T:X\to Y$ ...
Ioana Ghenciu's user avatar
2 votes
0 answers
43 views

Distributions and time-kernels

Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
G. Blaickner's user avatar
  • 1,429
-1 votes
0 answers
94 views

Why define Schwartz by supremum rather than limit?

The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite. However,...
Ponder Stibbons's user avatar
-1 votes
0 answers
53 views

convergence of convolution in Bochner space

I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$ let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
Alucard-o Ming's user avatar

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