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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Edyta Kania (University of Wrocław)

Hardy spaces for Bessel-Schrödinger operators

Let $K_t = exp(-\mathbf{L}t)$ be the semigroup generated by a Bessel-Schrödinger operator on $L^2((0,\infty), x^\alpha dx)$ given by $$ \mathbf{L}f(x) = -f''(x) - \frac{\alpha}{x} f'(x) + V(x) f(x),$$ where $V\in L^1_{loc}((0,\infty), x^\alpha dx)$ is a non-negative potential and $\alpha>0$. We say that $\,\,\,f\in L^1((0,\infty), x^\alpha dx)$ belongs to Hardy space $\mathcal{H}^1(\mathbf{L})$ if $$\| \sup_{t>0} |K_t f| \|_{L^1((0,\infty), x^\alpha dx)} <\infty.$$ Under certain assumptions on $V$ and $K_t$ we characterize the space $\mathcal{H}^1(\mathbf{L})$ by atomic decomposition of local type. Moreover, we provide an application of this result for any potential $0\leq V\in L^1_{loc}((0,\infty), x^\alpha dx)$ and for $\alpha\in(0,1)$.
The talk is based on joint work with Marcin Preisner (University of Wrocław).