Questions tagged [harmonic-analysis]
Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
1,421
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Extension of a Szegő Kernel to the boundary
Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.
Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...
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84
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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 $f\in L^1(B(0,1))$, $\alpha\geq 0$ :
$$
M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
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Urysohn's lemma for Bochner functions?
Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:
If $U$ is an open ...
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de Rham theorem for tempered distributions
I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
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Maximal function in Orlicz space
Consider the maximal operator defined for a function $f\in L^1_{loc}$:
$$
Mf : x\mapsto \sup_{r>0} \frac{1}{|B(x,r)|} \int\limits_{B(x,r)} f.
$$
It is well know that $M : L^1 \to L^{(1,\infty)}$ ...
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Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
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Does periodic pattern arise in syndetic pattern
We wonder for two "large" sets $I,J\subseteq \omega$, if $(J-J)\cap I=\emptyset$, then it must be due to certain periodic pattern.
We say $I\subseteq \omega$ is $t$-syndetic iff for every $n\...
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About weak integrals: Appendix of Folland's book "A course in abstract harmonic analysis"
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
All integrals are here to interpreted in the weak sense (see p285 in Folland's book). Why is ...
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157
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Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
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120
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Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
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Analytic approximation of the step function in $L^p$ norm
Motivation: Euler-Maclaurin formula uses calculus to estimate discrete sums. I wonder what one can do by reverse engineering. A concrete problem I ran into is the following.
Question: Let $\chi: \...
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73
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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 ...
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149
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An oscillatory integral
Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates?
\begin{...
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A closed ideal in $L^1(T)$
Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$.
Let $I=...
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Integral of $\ln(1/|f|)$ for $f$ bandlimited
I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...
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1
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466
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Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
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154
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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)=\...
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A Sobolev embedding theorem for functions on spheres
$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
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Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
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Estimates about prime numbers: a lemma in Bourgain's article
For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds:
\begin{equation}
|\{q\in \mathbb{N},q<Q: \text{all ...
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1
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149
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Finite Hindman theorem
Consider the following finite version Hindman theorem:
For every sufficiently large $N\in\omega$ and 2-partition of $N=N_0\cup N_1$, there are $i<2,a,b,c\in N_i$ such that $a+b=c$.
The only proof I ...
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The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
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How to show the solution map of NLS is not smooth?
Let $u(\delta, t)$ satisfy
$$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$
Note that the mapping:
$$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$
...
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Maximal ergodic inequality
A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$,
$$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \...
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Pointwise convergence of trigonometric series
$f$ is said to have trigonometric expansion if some series $\sum_{n\in\mathbb{Z}}c_ne^{inx}$ converges pointwise to $f(x)$. On the second page of the article Trigonometric series and set theory, ...
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Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
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Equivalent action of convolution of directional derivative
I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented.
Let $k*f(x)...
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Equality of two subharmonic functions
Let $u\leq v$ be two locally bounded subharmonic functions in a domain in $\mathbb{R}^n$. Assume that $u=v$ on a dense subset.
Is it true that $u=v$ everywhere?
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Stein's book on harmonic analysis
My background :
I am a Math PhD student. I will most probably work in harmonic analysis on Euclidean spaces. I am a fan of Folland's Real analysis and I have thoroughly studied first 8 chapters of ...
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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{...
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73
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A problem arising from Wiener-Levy theorem on the real line
Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
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Stability of Hajłasz-Sobolev class under post-composition
Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev?
Assumptions/Setup
Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
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67
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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 ...
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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 ...
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Is the cone of positive elements in $L^1(G)$ norm closed?
Let's consider $L^1(G)$, the Banach $*$- algebra of all Lebesgue integrable functions on the locally compact group $G$. Put $L^1(G)_+$ by the cone of positive elements given by $\{\sum_1^n f_i^**f_i: ...
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77
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Real life applications of distributions through models or simulations [closed]
What are the areas we can apply distributions in classical harmonic analysis? I don't mean probability distributions but distributions that are continuous linear functionals on the space of test ...
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1
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182
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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]...
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1
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452
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Unifying two definitions of $L^\infty$
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$.
Definitions:
A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
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0
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158
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Inequality on the dual space of $H^s$
Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ?
For instance, assume ...
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Probability measure on the boolean cube with small support and small Fourier transform
[Edit: added details on the Reed-Muller codes]
Are there explicit (non random) constructions of probability measures on $D_N = \{0,1\}^N$ with support of size $O(N)$ and with all nontrivial Fourier ...
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0
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150
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Erdős–Turán inequality for complex numbers
Consider the following set of complex numbers in the upper half plane: $$\{ic_n \pm \gamma_n: 0 \leq n \leq N, \hspace{1 mm} c_0=\gamma_0=0, c_n,\hspace{1 mm} \gamma_n>0\}.$$
Assume that this set ...
- 507
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1
answer
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Fourier multipliers and transference on cyclic groups
It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\...
2
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0
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130
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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 ...
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0
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150
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On the best constant for Carleson's embedding theorem
In "Interpolations by bounded analytic functions and the corona problem", Carleson introduced Carleson measures (for Hardy spaces) and proved the famous embedding theorem according to which $...
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votes
0
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146
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Non-commutative harmonic analysis on the discrete Heisenberg group
Question: Is there a linear map $\mathcal F$ from the Hilbert space of $\ell^2$ functions on the discrete Heisenberg group to some Hilbert space of functions $ L^2(\bigcup \{\Omega_n\}) $, such that:...
- 9,421
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votes
0
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202
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Regularity of $|u|^{\alpha}$ when $u$ is Schwartz
Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
3
votes
0
answers
367
views
Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to
\begin{equation}
\begin{cases}
-\Delta u=0 \quad &\mbox{in $\Omega$}\\
\frac{\partial ...
- 1,320
1
vote
1
answer
178
views
Is $\log|h|$ BMO when $h$ is analytic?
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. A function $f$ is called a bounded mean oscillation (BMO for short) function on $\mathbb{D}$ if
$$\sup_{disc\,\subset\, \mathbb{D}}\dfrac{1}{|...
- 11
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votes
1
answer
507
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Does the following version of the Coifman–Meyer Theorem exist?
Recall the Coifman–Meyer Theorem as stated in Grafakos and Oh - The Kato-Ponce Inequality.
Theorem: Let $m \in L^{\infty}\left(\mathbf{R}^{2 n}\right)$ be smooth away from the origin and satisfy
$$
\...
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0
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143
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barrier functions ; harmonic functions; gradient estimates (textbook reference)
Suppose $\Omega$ is a bounded domain in Euclidean space and you can take it as convex as you wish. Is there a good place to look for a bunch of barrier type functions.
In particular I am looking for ...
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