All Questions
11 questions
1
vote
2
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152
views
Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
- 3,625
6
votes
1
answer
193
views
Oscillatory integrals with a decaying factor in the integrand
Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased):
Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...
- 63
1
vote
1
answer
135
views
Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$
Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that
$$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$
and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
- 123
23
votes
1
answer
528
views
A characterization of constant functions
In How to recognize constant functions. Connections with Sobolev spaces (Russian Math Surveys 57 (2002); MSN), H. Brezis recalls the following fact:
Let $\Omega\subset{\mathbb R}^N$ be connected ...
- 52.3k
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405
1
vote
1
answer
222
views
Multidimensional improper Riemann integrals with oscillatory kernels: Existence
I have asked this question three weeks ago here
https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930
but received no relevant answers.
Let $n\geq 2$ ...
- 143
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
3
votes
1
answer
93
views
Does $p$ integrability in n-1 dimensions give higher integrability in $n$ dimensions?
Restrict everything to a ball in $n$ dimensions, let $x$ represent the first $n-1$ variables, and $t$ the $n-$th variable. It is obvious by Holder's Inequality that
$$
\int\limits_t\left(\int\limits_x|...
- 391
15
votes
1
answer
602
views
Integrability property of polynomials in several variables
This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...
- 151
6
votes
1
answer
741
views
Is the following integral nonzero?
Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...
- 1,997
3
votes
1
answer
480
views
To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$
I wants to understand the integrals of the form
$$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$
where $\mu(\alpha)$ is a non decreasing function such ...
- 1,051