We use microlocal and paradifferential techniques to obtain$L$^{8}norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal$L$^{$q$}bounds, in the range 2⩽q⩽∞, for$L$^{2}- normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp$L$^{$q$}estimates in higher dimensions for a range of exponents q̅_{n}⩽$q$⩽∞.

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Let $\mathcal{D}$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function$f$∈ $\mathcal{D}$ to be$cyclic$, i.e. for {$pf$:$p$is a polynomial} to be dense in $\mathcal{D}$ .

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