All Questions
10,935 questions
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45
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Characterization of the Picard's condition for integral equation
Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
- 143
1
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0
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105
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Applications of finite speed of propagation property
Consider the Laplace operator $\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^n$. Let $E_{\lambda}$ be the spectral resolution of $\Delta$, and
$$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
- 287
2
votes
0
answers
104
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Clique-coclique and uncertainty
The clique-coclique inequality states that for a graph $G$ on $n$ vertices that is either distance-regular or vertex-transitive, the independence number $\alpha(G)$ and the clique number $\omega(G)$ ...
- 23k
2
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1
answer
69
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Spectral threshold effect: examples
I know that the effect of homogenization can be treated as a spectral threshold effect. I want to know more examples of spectral threshold effects in mathematical physics.
16
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2
answers
1k
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Examples of Banach manifolds with function spaces as tangent spaces
I have recently been learning the theory of Banach manifolds through Serge Lang's book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the ...
- 161
1
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1
answer
242
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near ergodic theory question
Let $(X,\sigma,\mu)$ be a measure space, $\mu(X)=1$. End let $T:X\to X$ be a measurable map such that
$$D\in\sigma\Longrightarrow \mu(T^{-1}D)=\mu(D).$$ Let $f:X\to\mathbb{R}_+$ be an integrable ...
2
votes
1
answer
251
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Log-Sobolev constant
Let $\nu \propto e^{-f}$ be a probability density on $\mathbb{R}^d$ with full support. We say $\nu$ satisfies the log-Sobolev inequality (LSI) with constant $\alpha$ if for every smooth function $g:\...
- 519
1
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1
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88
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Mean values of polynomial and holomorphic matrices
Lemma. Assume $H: \mathbb{R} \to \mathbb{R}^{d \times d}$ is a polynomial of degree $m$, such that for all $x \in \mathbb{R}$, $H(x)$ is a symmetric semidefinite matrix. For all $n \geq 0$ and real ...
- 981
3
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0
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126
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$L^\infty-L^1$ norm of an oscillatory integral operator
Let $\lambda>1$, $x,y\in [0,1]$, and $K_\lambda(x,y)=(1+\lambda|x-y|)^{-2}e^{i\lambda|x-y|}$. Consider the operator
$$T_\lambda f(x)=\int_0^1 K_\lambda(x,y)f(y)dy.$$
We want to precisely estimate ...
- 31
0
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1
answer
241
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Norm functions induced by convex bodies
Given a centrally symmetric convex body $K$ in the plane (with smooth boundary), it is easy to see that there exists a norm function $g:\mathbb{R}^2\to \mathbb{R}_{\geq 0}$ for which $K$ is the unit ...
- 191
8
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1
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723
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Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials
Let $ X_N = \text{span} \{\cos(2\pi lx): l=0, \cdots, N-1 \} $ with $ x \in [0, 1] $ and $ Y_N = \{v =(v_0, \cdots, v_{N-1}): v_j \in \mathbb{C}\} = \mathbb{C}^N $. Then $ X_N $ is the space of ...
- 105
2
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1
answer
199
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Gaussian Poincare inequality in $1$ dimensions together with localization issue
Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$.
Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$.
Then, for any smooth mapping $f : \...
- 3,477
2
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0
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62
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Continuous-time Wold decomposition
I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line.
I am aware of the classic result in the book from Rozanov, which ...
- 21
4
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1
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267
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Is $T$ totally bounded when $C_u(T)$ is separable?
I'm seeking help with a question regarding the space of bounded and uniformly continuous functions $C_u(T,d)$, where $(T,d)$ is a metric space. In this context, $C_u(T)$ is a closed subspace of $C_b(T)...
- 235
2
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41
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Blow up for certain classes of distributions
Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
- 4,153
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2
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125
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Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 825
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0
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138
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Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 791
4
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1
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418
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Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u_t+uu_x=u_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 43.2k
1
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1
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117
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Bounds of periodic functions formed from infinite series of shifts
Recently, I have become quite obsessed with the follow series:
$$f(t,x)=\sum_{m=-\infty}^{\infty} (-1)^m f(t+x m)$$
where $f$ is analytic. This series automatically produces a periodic function with ...
- 402
2
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1
answer
99
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Definite negative functions and length functions
$\DeclareMathOperator\ND{ND}$I am reading E. Bedos paper on heat properties for groups.
Let's denote, for a group G, $$\ND^+_0(G) := \{d : G \to [0,+\infty[\; : \;d \text{ is negative definite and }d(...
- 21
3
votes
3
answers
777
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Radon-Nikodym property for space of signed measures
Given a measurable space, the vector space of signed measures is a Banach space. Does it have the Radon-Nikodym property? What if the space is of a special type, such as a nice topological space with ...
- 2,247
5
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1
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216
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Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 4,153
2
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1
answer
136
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Eigenvectors of the dual of positive irreducible operators
This question was previously posted on MSE.
Let $E$ be a Banach lattice such that $E$ is an $M$-space. Assume that $T\colon E\to E$ is a positive bounded non-compact irreducible linear operator with ...
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7
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3
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935
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Elegant proofs of $\bar{\partial}z^{-1} = 2\pi \delta_0$
For a function $f(x,y)$ on $\mathbb{R}^2,$ defined possibly outside the origin, write
$$\int_\epsilon ' f \,dx\,dy : = \int_{\mathbb{R}^2\setminus D_\epsilon}f \, dx\,dy,$$
(the integral on the ...
- 7,962
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1
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241
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What is convergence in distribution of random variables taking values in a non-metrizable product space?
Let $F = (F(x) : x \in \mathbf{R}^n)$ be a family of $\mathbf{R}^k$-valued random variables indexed by $\mathbf{R}^n$ (to be clear there is a single probability space $(\Omega,\Sigma,\mathbf{P})$ such ...
- 1,179
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145
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Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
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224
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On the conditions of convergence in the generalized Riemann-Lebesgue lemma
I am reposting the following question that I asked in the MSE site here.
As I mentioned there, The following generalizations of the Riemann-Lebesgue lemma are rather well known (Kahane, C. S., ...
- 271
2
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1
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471
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Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?
Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123.
Why is it possible to choose an ...
- 175
2
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2
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281
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Most general reverse Hölder inequality for polynomials
Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$,
$$\|p\|_{L^\infty(a,b)} \...
- 981
5
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1
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1k
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Chain rule in Sobolev space
In the theory of Sobolev space, we have the following chain rule:
For a uniformly Lipschitz function $F : \mathbf{R}\to \mathbf{R}$ such that $F(0)=0$,
and $u\in W^{1,1}(\mathbf{R}^n)$, then we have ...
- 705
44
votes
10
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47k
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Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
4
votes
1
answer
398
views
Inequality for Fourier transform of a power exponential function
Let
$$
f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }},
x \in \mathbb{R}, 0<\alpha<2,
$$
where
$\phi_1(\alpha)=\frac{\alpha}{2}
\left\{{\{\Gamma(3/\alpha)\}^{1/...
- 178
1
vote
1
answer
354
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Mach's principle, Newton's law and Hilbert sphere?
(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.)
I wanted to share with you something I stumbled upon ...
- 3,074
5
votes
1
answer
177
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Sobolev inequality with holes
Classical Sobolev inequality says, $n\geq 3$, we have
\begin{equation}
\left(\int_{\mathbb{R}^n}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C(n)\left(\int_{\mathbb{R}^n}|\nabla u|^{2}\right)^...
- 471
10
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1
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474
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A basic estimate of exponential sums
Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:
\begin{equation}
\sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
- 351
6
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3
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415
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Examples of amenable Banach algebras which have non-amenable subalgebra
I am looking for examples of amenable Banach algebras which have non-amenable subalgebra
I know
1: Each amenable Banach algebra has a bounded approximate identity
2: If $I$ be a closed ideal in an ...
- 823
1
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1
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160
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On an integral equation
Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...
- 4,153
0
votes
1
answer
79
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Convergence in sequential Lebesgue spaces
Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
- 1,101
2
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0
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126
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Differential equations: trying to connect a nonlinear equation to a linear one
The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
- 383
1
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0
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111
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Residues of analytic operators
Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ ...
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118
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For any smooth function $f$ on $[0,1]$, do we have $\lVert P_N f \lVert_{1} \leq \lVert f \rVert_1$ for $P_N$ defined by Fourier expansion?
Let $C^\infty[0,1]$ be the space of periodic smooth functions on $\mathbb{R}$ with the period $1$. It is well-known to be a Frechet space with the uniform convergence of all derivatives.
Also, $\{ e^{...
- 3,477
0
votes
1
answer
185
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Spectrum of a product of a symmetric positive definite matrix and a positive definite operator
Let $\mathbf H$ be an infinite dimensional Hilbert space.
I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
- 63
0
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0
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80
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Continuity of linear map on tensor product spaces with different norm properties
I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...
- 177
6
votes
1
answer
251
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Finite representability of $\ell_p$ in subspaces of $L_p(0,1)$
Let $M$ be a closed subspace of $L_p(0,1)$, $1<p<\infty$, $p\neq 2$.
Suppose that M contains copies of $\ell_p^n$ uniformly.
Does $M$ contain a copy of $\ell_p$?
The result is true for $p=1$, ...
- 4,461
21
votes
1
answer
1k
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(update) Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
Problem: Given three positive integers $0 < n_1 < n_2 < n_3$ such that
$$n_1 + n_2 \ne n_3, \quad n_2 \ne 2n_1, \quad n_3 \ne 2n_1, \quad n_3 \ne 2n_2,$$
is there always a real number $x$ ...
- 1,053
4
votes
0
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495
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Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
- 175
6
votes
1
answer
679
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Fourier optimization problem related to the Prime Number Theorem
Let $\kappa>0$ be given. What is the function
$f:\mathbb{R}\to [0,\infty)$ with $\int_\mathbb{R} f(x) dx = 1$ such that
$$\int_\mathbb{R} |x| f(x) dx + \kappa \int_{|t|\geq T}\left| \frac{\widehat{...
- 20.2k
4
votes
2
answers
392
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 5,380
1
vote
2
answers
213
views
How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$
Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that
\begin{equation}\tag{1}\label{1}
\int_a^b \...
4
votes
2
answers
315
views
Is this an $L^p-L^{\infty}$ operator?
Let $1\leq p <\infty$ and let $p^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions:
$$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t}
\int_{|x-...
- 852