Weighted-integrable $H^p$ functions have vanishing moments in the atomic range

Smoothness of $\widehat{f}$ at the origin. For each coordinate, the exponential difference quotient is bounded by $2\pi \mathrm{\mid }{x}_j\mathrm{\mid }$, since $\mathrm{\mid }{e}^{iu}-1\mathrm{\mid }\le \mathrm{\mid }u\mathrm{\mid }$. Iterating dominated convergence with the assumed integrable functions $\mathrm{\mid }{x}^{\alpha }f\mathrm{\mid }$ proves the derivative formula in [F2]; dominated convergence applied to each derivative integrand proves its continuity. Since $x^{\alpha }f\in L^1$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le s$, [F2] gives that $\widehat{f}$ is $s$ times continuously differentiable near the origin and that ${\mathrm{\partial }}^{\alpha }\widehat{f}(0)$ is the Fourier transform of $(-2\pi ix{)}^{\alpha }f$ at the origin.

A nonvanishing lowest derivative contradicts the little-$o$ decay. Suppose some ${\mathrm{\partial }}^{\alpha }\widehat{f}(0)\mathrm{\ne }0$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le s$, and choose such an $\alpha$ of minimal total degree $m$. Put $\gamma =n(1\mathrm{/}p-1)\ge 0$. If $m=0$, then $\widehat{f}(0)\mathrm{\ne }0$; choose any unit vector $\eta$. Continuity from [F2] gives $\mathrm{\mid }\widehat{f}(t\eta )\mathrm{\mid }\ge \mathrm{\mid }\widehat{f}(0)\mathrm{\mid }\mathrm{/}2$ for all sufficiently small $t>0$, contradicting [F1], which says $\widehat{f}(t\eta )=o(t^{\gamma })$ and hence tends to zero. If $m\ge 1$, every derivative of order below $m$ vanishes. By [F2], $\widehat{f}$ is $C^m$ near $0$, so [F3] applied to its real and imaginary parts gives, for fixed $\eta \in {\mathbb{R}}^n$, $\widehat{f}(t\eta )=t^{m}Q(\eta )+o(t^m)(t\downarrow 0),Q(\eta )={\sum }_{\mathrm{\mid }\beta \mathrm{\mid }=m}\frac{{\mathrm{\partial }}^{\beta }\widehat{f}(0)}{\beta !}{\eta }^{\beta }\mathrm{.}$ This complex homogeneous polynomial is not identically zero, so choose a unit vector $\eta$ with $Q(\eta )\mathrm{\ne }0$. Then $\mathrm{\mid }\widehat{f}(t\eta )\mathrm{\mid }\ge c{t}^m$ for all sufficiently small $t>0$. Since $m\le s=\lfloor \gamma \rfloor \le \gamma$, one has $t^m\ge t^{\gamma }$ for $0<t\le 1$, contradicting [F1]. Thus every ${\mathrm{\partial }}^{\alpha }\widehat{f}(0)$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le s$ vanishes.

Conclusion. By [F2], ${\mathrm{\partial }}^{\alpha }\widehat{f}(0)=(-2\pi i{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}\int x^{\alpha }f$ for every $\mathrm{\mid }\alpha \mathrm{\mid }\le s$; step 2.1 shows these derivatives all vanish, so $\int x^{\alpha }f=0$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le s$. For $p=1$ one has $s=0$, so the integral of $f$ vanishes; applying this to a compactly supported $L^1$ function $f\in H^1$ gives $\int f=0$, and a compactly supported $L^1$ function with $\int f\mathrm{\ne }0$ cannot be in $H^1$.

Remark on the hypothesis. For $p=1$ every $H^1$ function that is a locally integrable function automatically has $f\in L^1$ by [F4], so the "compactly supported $L^1$" formulation is a special case; for $p<1$ the hypothesis $x^{\alpha }f\in L^1$ is a genuine additional assumption. This corollary proves the stated vanishing moments and no more.