Articles of hardy spaces

Suppose $\phi$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$. Then $\phi\in W^{2,1}$

I’m trying to prove the statement in the title in as simple a way as possible. It is Theorem 3.2.9 in Helein’s book “Harmonic maps, conservation laws, and moving frames”, although it is not proved there. The statement is as follows. Suppose $\phi\in\mathbb{R}^m$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$, where […]

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} \end{equation} and a locally integrable function $f$ on $\mathbb{R}^n$, we define the grand maximal function of $f$ as \begin{equation} f^{\ast}(x)\equiv \sup_{t>0}\sup_{\phi\in\mathcal{T}}|\phi_t\ast f(x)|\equiv\left|\int_{\mathbb{R}^n}\frac{1}{t^n}\phi\left(\frac{x-y}{t}\right)f(y)\ \mathrm{d}y\right|\quad(x\in\mathbb{R}^n). \end{equation} The Hardy-Littlewood maximal function is defined as \begin{equation} Mf(x)\equiv\sup_{t>0}\frac{1}{|B(x, t)|}\int_{B(x, t)}|f(y)|\ \mathrm{d}y\quad […]

Integrability of Maximal Convolution Operator

Let $f\in L^{\infty}(\mathbb{R}^{n})$ be supported in the unit ball and have mean zero. Let $\phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n})$ be a Holder continuous function with exponent $\alpha>0$. Define an approximation to the identity $\left\{\phi_{t}\right\}_{t>0}$ by $\phi_{t}:=t^{-n}\phi(\cdot/t)$, and consider the maximal convolution operator $M_{\phi}$ defined by $$M_{\phi}(f)(x)=\sup_{t>0}\left|(f\ast\phi_{t})(x)\right|, \qquad\forall x\in\mathbb{R}^{n} \tag{1}$$ If $\phi$ is compactly supported in a ball […]

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 […]

Criterion for Membership in Hardy Space $H^{1}(\mathbb{R}^{n})$

Let $H^{1}(\mathbb{R}^{n})$ denote the real Hardy space (I am agnostic about the choice of characterization). It is known that if $f:\mathbb{R}^{n}\rightarrow\mathbb{C}$ is a compactly supported function (say in a ball $B$) such that $\int f=0$ and $$\int_{\mathbb{R}^{n}}|f(x)|\log^{+}|f(x)|dx<\infty$$ then $f\in H^{1}(\mathbb{R}^{n})$ and moreover, we have an estimate of the sort $$\|f\|_{H^{1}}\lesssim |B|\|f\|_{L\log L(dx/|B|)}$$ See Lemma 3.10 […]