All Questions
Tagged with real-analysis harmonic-analysis
274 questions
2
votes
0
answers
75
views
On Dirichlet eigenfunctions of a domain
Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ ...
- 4,135
7
votes
1
answer
370
views
Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 71
5
votes
3
answers
627
views
If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?
Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that
$$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$
Do we have $\|f\...
- 517
-1
votes
1
answer
213
views
Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
- 581
5
votes
0
answers
135
views
Quantifying the degree of continuity of a function via perturbations
Let $f: \mathbb R \to \mathbb R$ be a measurable function. Define the perturbation operator $T_f$ on measurable functions $g: \mathbb R \to \mathbb R$ by
$$T_f (g)(x) := f(x + g(x)) - f(x).$$
Observe ...
- 6,195
2
votes
1
answer
60
views
Specific estimation of the norm for a linearly transformed function in $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$
According to the standard definition, $\mathcal{S}_0^{\beta}(\mathbb{R})$ is a subspace of smooth functions on $\mathbb{R}$ with the property that
\begin{equation}
\lvert x^k f^{(q)}(x) \rvert \leq CA^...
- 3,477
2
votes
1
answer
174
views
Description of $\mathcal{S}^{2}_0(\mathbb{R})$ and certain class of functions inside it
I am currently reading volume 2 of "Generalized Functions" by Gelfand and $\mathcal{S}^{2}_0(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such ...
- 3,477
0
votes
1
answer
143
views
Existence of smooth functions $f$ satisfying $\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq C B^q k^{1/8} q^{q/2}$
$\mathcal{S}^{1/2}_{1/2}(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that
\begin{equation}
\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq ...
- 3,477
17
votes
3
answers
975
views
Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds
For real variable $x$, the function
\begin{equation}
f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n
\end{equation}
clearly has infinite radius of convergence and defines a $C^\infty$ function on $\...
- 3,477
5
votes
0
answers
132
views
Regularity of the spherical mean of a compactly-supported function
The problem
Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$.
Then, consider ...
- 555
2
votes
1
answer
157
views
Inequality with decreasing rearrangement and non-decreasing function
This question is a continuation of the question here.
Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$...
- 459
4
votes
1
answer
367
views
Inequality with decreasing rearrangement function
Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive ...
- 459
8
votes
1
answer
461
views
On critical points of harmonic functions
Let $u \in C^{\infty}(\mathbb R^3)$ be harmonic. Suppose that $u$ has no critical points outside the unit ball but that it has at least one critical point inside the unit ball.
Does it follow that $u$ ...
- 4,135
7
votes
2
answers
508
views
Why is $\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2})$ harmonic if $u$ is harmonic?
I found myself trying to prove the following, but I had to compute everything explicitly.
It is well known that if $u:\mathbb{R}^n\to\mathbb{R}$ is an harmonic function on $\mathbb{R}^n$, then the so-...
- 213
2
votes
0
answers
182
views
Smooth function supported on a "short" interval whose derivatives are $L^1$ bounded?
Let $P$ be a large number. Let $[a,b]$ be a fixed interval. Is it possible to construct a function $w$ with the following properties?: $w$ is smooth,
the support of $w$ is contained in $I = [a- 1/P, b ...
- 3,625
1
vote
1
answer
186
views
Existence of a smooth function that approximates a characteristic function of an interval with certain property
Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the ...
- 3,625
3
votes
1
answer
202
views
An inequality equivalent to Hörmander's condition $\sup_{y\in\mathbb R^n}\int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx<\infty$
This problem has been asked in MSE, but got no answers. I guess that this exam problem may be a small lemma in some research papers, so I post it here on MathOverflow.
Let $K\in L_{\text{loc}}^1(\...
- 517
4
votes
1
answer
661
views
The decay of Fourier coefficients and the continuity of functions
Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not ...
- 1,219
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
4
votes
1
answer
355
views
Sharpest version of semiclassical Calderon-Vaillancourt theorem
Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
- 854
3
votes
1
answer
157
views
Weak convergence of integral averages
Note: This is a refinement of a previous problem.
Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f_n$ by
$$f_n ...
- 6,195
6
votes
1
answer
313
views
Convergence of integral averages in $L^1$
Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f_n$ by
$$f_n (x) := \frac{1}{2g_n (x)} \int_{x - g_n (x)}^{x + ...
- 6,195
2
votes
0
answers
216
views
Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
6
votes
0
answers
213
views
Equivalent forms of Fourier restriction conjecture
this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow.
I'm reading Pertti Maattila's book ...
- 196
3
votes
1
answer
259
views
A kind of vector-valued Littlewood–Paley inequality for arbitrary intervals
The title may be inappropriate, and I apologize for that.
I'm writing a reading report on my harmonic analysis course. My topic is the Littlewood-Paley inequality for arbitrary intervals, which was ...
- 517
2
votes
0
answers
87
views
Maximal function to high power
Consider the following maximal function : in dimension $n$ consider $B(0,1)\subset \mathbb{R}^n$ the unit ball, if $f\in L^1(B(0,1))$, $\alpha\geq 0$ :
$$
M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
- 363
4
votes
0
answers
82
views
Maximal function in Orlicz space
Consider the maximal operator defined for a function $f\in L^1_{loc}$:
$$
Mf : x\mapsto \sup_{r>0} \frac{1}{|B(x,r)|} \int\limits_{B(x,r)} f.
$$
It is well know that $M : L^1 \to L^{(1,\infty)}$ ...
- 363
1
vote
2
answers
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.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
- 3,625
2
votes
0
answers
152
views
Riesz transform of constant function
My one-line question would be, what is the Riesz transform of the constant function, identically equal to 1 on $\mathbb{R}^2$?
But more fundamentally, my question stems from some confusion about the ...
- 287
2
votes
1
answer
155
views
Exponential integrability of a sum of approximations of disjoint intervals characteristic functions
Let $I=[0,1]$ be the unit segment, and let $(I_n)_{1\leq n \leq N}$ be $N$ almost disjoint sub-intervals $I_n=[t_n-\delta_n,t_n+\delta_n]$ of $I$ (that is, their interior are disjoint). Let $\chi(x)=\...
- 489
9
votes
2
answers
1k
views
Density of restrictions of harmonic functions inside a ball
Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let
$$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
- 4,135
0
votes
0
answers
75
views
Extracting the point mass measure of some type of positive measures
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals.
Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
- 4,058
0
votes
1
answer
88
views
An equation in the convolution measure algebra on reals
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals.
Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
- 4,058
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
3
votes
0
answers
137
views
Is there a characterization of tempered functions whose Fourier transforms are also tempered functions?
A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions.
This MSE question asked ...
- 315
8
votes
1
answer
2k
views
Summary of sufficient conditions for convergence of Fourier series
I would like to summarize various sufficient conditions for various modes of convergence of Fourier series. The followings are what I have gathered so far:
$L^p$ convergence:
if $f \in L^p(\mathbb{T}...
- 315
0
votes
0
answers
84
views
Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
3
votes
1
answer
148
views
How to compute the left-inverse of $f$ in the sense of $\approx$? [duplicate]
For two functions $\varphi,\psi:\Omega\times[0,\infty)\to[0,\infty)$, $\varphi\approx\psi$ means that there exist constants $c_1,c_2>0$, such that $\forall t\geq0$, $\forall x\in\Omega$, $$c_1\...
- 51
2
votes
2
answers
243
views
Given a specific function $f$, how to compute the left-inverse of $f$ in the sense of $\approx$?
For a non-negative function $\varphi$ defined on $[0,\infty)$, the left-inverse $\varphi^{-1}$ of $\varphi$ is defined by setting, $\forall t\geq 0$,
$$\varphi^{-1}(t):=\inf\{u\geq0:\varphi(u)\geq t\}....
- 51
7
votes
2
answers
997
views
$L^p$ bounds on tails of bounded $L^q$ sequences
Note: This is a generalisation of an earlier problem as suggested by user Jochen Glueck in the comments.
Let $1 \leq p < q \leq \infty$, and $f_n: [0, 1] \to \mathbb R$ be a sequence of functions ...
- 6,195
1
vote
2
answers
140
views
Extending a discrete singular kernel
Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
- 1,879
1
vote
1
answer
518
views
On level sets of smooth functions in a bounded domain
Let $\Omega$ be a bounded domain in $\mathbb R^n$, $n\geq 2$, with a smooth boundary and let $f$ be a smooth function on $\bar\Omega$. Is there a natural condition that one can impose on $f$ ( say in ...
- 11
8
votes
2
answers
773
views
Points where harmonic functions fail to give a coordinates system
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
- 4,135
9
votes
0
answers
347
views
Can one prove Rademacher’s theorem via the rising sun lemma?
The classical Rademacher’s theorem states that Lipschitz continuous functions on $\mathbb R^n$ are differentiable almost everywhere.
In dimension one, a stronger result holds - it can be shown that ...
- 6,195
1
vote
0
answers
142
views
Fourier transform of the Bochner-Riesz multipliers
How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define:
$$
\hat{m_{\lambda}}(x)=\int\limits_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \...
- 351
6
votes
1
answer
397
views
Absolute values of two functions and absolute values of their Fourier transform coincides
Let $f, g \in L^2(\mathbb{R})$.
Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?
I am not able to prove it or ...
- 489
0
votes
0
answers
132
views
Integral kernel of resolvent of Sub-Laplacian?
Consider the Laplacian $-\Delta: C^\infty (S^1) \to C^\infty(S^1)$ where $\mathbb R/\mathbb Z=S^1$ and for a periodic function $f:\mathbb R\to\mathbb R$ we have $-\Delta f=-f''$.
For the orthonormal ...
8
votes
2
answers
622
views
Vanishing rate of a harmonic function near a boundary point
Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is,
$$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$
for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
- 903
2
votes
0
answers
164
views
(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis
It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
- 1,245
2
votes
0
answers
58
views
Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
- 4,135