On Hardy spaces associated with Bessel operators
The authors consider several Hardy spaces connected with the operators \[ \begin{aligned}\Delta_{\lambda}f(x)&=-\frac{d^2}{dx^2}f(x)-\frac{2\lambda}{x}\frac{d}{dx}f(x), \quad x >0,\quad\text{and} \\ S_{\lambda}f(x)&=-\frac{d^2}{dx^2}f(x)+\frac{\lambda^2-\lambda}{x^2}f(x), \quad x > 0.\end{aligned} \] Both operators generate the Riesz transform, the heat semigroup and the Poisson semigroup. These operators are denoted by \(R_L\), \(e^{-tL}\) and \(e^{-t\sqrt{L}}\), where \(L=\Delta_{\lambda}\) or \(L=S_{\lambda}\). Then \(H^1_{\text{Riesz}}(L)= \{f\in L^1:\|f\|_{H^1_{\text{Riesz}}(L)}=\|f\|_{L^1}+\|R_L\|_{L^1} < \infty\}\) and \(H^1_{\max}(L)=\{f\in L^1:\|f\|_{H^1_{\max}(L)}=\|f\|_{L^1}+\|\sup_{t > 0}|e^{-t\sqrt{L}}|\|_{L^1} < \infty\}\). Let \(\rho(x,y)=|x-y|\), \(dm_{\lambda}(x)=x^{2\lambda}\,dx\). A measurable function \(a(x)\) is an \(H^1_{\text{cw}}\)-atom, if there exists a bounded interval \(I\subset [0,\infty)\) such that supp\((a)\subset I\), \(\|a\|_{L^{\infty}(\mathbb R_+,dm_{\lambda})}\leq 1/m_{\lambda}(I)\) and \(\int^{\infty}_0a(x)\,dm_{\lambda}(x)=0\). A function \(f\in L^1(\mathbb R_+,dm_{\lambda})\) belongs to \(H^1_{\text{cw}}(\mathbb R_+,dm_{\lambda})\) iff \(f=\sum^{\infty}_{j=1}\alpha_ja_j\) in \(L^1(\mathbb R_+,dm_{\lambda})\), where \(a_j\) are \(H^1_{\text{cw}}\)-atoms and \(\sum^{\infty}_{j=1}|\alpha_j| < \infty\). A measurable function \(a(x)\) on \(\mathbb R_+\) is an \(F\)-atom, if (a) \(a=\delta^{-1}\chi_{(0,\delta)}\) or (b) there exists a bounded interval \(I\subset (0,+\infty)\), such that supp\((a)\subset I\), \(\int_Ia(x)\,dx=0\) and \(\|a\|_{L^{\infty}(\mathbb R_+)}\leq |I|^{-1}\). Here \(|I|\) denotes the length of \(I\) and \(\chi_E\) denotes the indicator of \(E\). A function \(f\in L^1(\mathbb R_+)\) belongs to \(H^1_F(\mathbb R_+)\) iff \(f(x)=\sum^{\infty}_{j=1}\alpha_ja_j(x)\), where the \(a_j\) are \(F\)-atoms and \(\sum^{\infty}_{j=1}|\alpha_j| < \infty\). Two main results of the paper are Theorem 1.7. Let \(\lambda >0\) and \(f\in L^1(\mathbb R_+,dm_{\lambda})\). The following assertions are equivalent: (i) \(f\in H^1_{\text{cw}}(\mathbb R_+,dm_{\lambda})\); (ii) \(f\in H^1_{\max}(\Delta_{\lambda})\); (iii) \(f\in H^1_{\text{Riesz}}(\Delta_{\lambda})\). Theorem 1.10. Let \(\lambda >0\). The following assertions are equivalent: (i) \(f\in H^1_F(\mathbb R_+)\); (ii) \(f\in H^1_{\max}(S_{\lambda})\); (iii) \(f\in H^1_{\text{Riesz}}(S_{\lambda})\). In both theorems the corresponding norms are equivalent.
scientific article; zbMATH DE number 5646617
scientific article