Articles of harmonic analysis

example of maximal operator that is integrable

We know that there are no nonzero functions $f \in L^1(\mathbb R^n)$ such that $Mf \in L^1(\mathbb R^n)$, where $Mf$ is the Hardy Littlewood maximal function. Can we find a maximal operator that is integrable for nonzero functions? More precisely, I would really appreciate some help with the following: Let $ \phi \in C^\alpha (\mathbb […]

Hardy Space Cancellation Condition

I have been reading the chapter on Hardy spaces in Stein’s Harmonic Analysis book, and I am having a lot of trouble figuring something out. The setting here is $\mathbb{R}^n.$ Let $f \in L^q$ be compactly supported, and let $\phi$ be a Schwartz function. As usual, define $\phi_t(x) = t^{-n} \phi(t^{-1} x).$ Define the maximal […]

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family $\{E_{i}\}$ of pairwise disjoint Borel sets of $\mathbb R$), and for which $\mu(E)$ is finite if the closure of $E$ is […]

Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has sides parallel to axis) is defined as the following integral: $\frac{1}{|Q|}\int_{Q}|u(y)-u_Q|\,\mathrm{d}y$, where $|Q|$ is the volume of $Q$, i.e. its Lebesgue measure, $u_Q$ is the […]

A bound for the product of two functions in BMO

The question: Let’s consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I’m trying to figure out if the following inequality is true $$ \|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}. $$ My approach: I tried using the duality with $H^1$. I get $$ \|fg\|_{BMO}\leq \sup_{\|h\|_{H^1}\leq 1} 2\|f\|_{L^\infty}\|g\|_{BMO}\||h|\|_{H^1}. $$ So, if in the periodic case we have $h\in H^1\rightarrow |h|\in H^1$, I’m done. […]

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: $$\int_{-\infty}^{+\infty}e^{it\lambda}\mathrm{d}\mu(\lambda)=0\implies\mu=0$$ How can I prove this?

Pontryagin dual of the unit circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? Any notes or suggestions will be appreciated.

Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that …the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is continuously embedded in $W^{1+\beta,1}(\mathbb{R}^{n})$ for each $\beta\in(0,1)$. Therefore, if $u$ and $\Delta u$ are in $L^{1}(\mathbb{R}^{n})$, then any first order derivative $D_{i}u$, $i=1,\ldots,n$, belongs to $W^{\beta,1}(\mathbb{R}^{n})$for each $\beta$, and […]

Haar measure scalars, what am I doing wrong here?

Let $\mu$ be a Borel measure on a topological space $X$, and let $E \subseteq X$ be Borel. Let $\phi$ be a homeomorphism of $X$, and let $\lambda$ be the measure given by $\lambda(A) = \mu(\phi(A))$. If $f: E \rightarrow \mathbb{C}$ is measurable and integrable, then so is $f \circ \phi: \phi^{-1}E \rightarrow \mathbb{C}$, and […]

Function invariant under Hilbert transform

Let $f\in C_0^\infty(\mathbb{R}^n)$ and $$Hf(x)= \operatorname{p.v.}\int_{\mathbb{R}}\frac{f(x-y)}{y} \, dx$$ the Hilbert transform of $f$. Is it possible that $Hf=f$ (a.e. and possibly after extending $H$ to $L^p$ space) ?