Schatten classes of generalized Hilbert operators

• Autores: José Angel Peláez Márquez , Daniel Seco
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 1, 2018, págs. 83-105
• Idioma: inglés
• DOI: 10.1007/s13348-017-0195-5
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let \mathcal {D}_v denote the Dirichlet type space in the unit disc induced by a radial weight v for which \widehat{v}(r)=\int _r^1 v(s)\,\text {d}s satisfies the doubling property \int _r^1 v(s)\,\text {d}s\le C \int _{\frac{1+r}{2}}^1 v(s)\,\text {d}s. In this paper, we characterize the Schatten classes S_p(\mathcal {D}_v) of the generalized Hilbert operators \begin{aligned} \mathcal {H}_g(f)(z)=\int _0^1f(t)g'(tz)\,\text {d}t \end{aligned} acting on \mathcal {D}_v, where v satisfies certain Muckenhoupt type conditions. For p\ge 1, it is proved that \mathcal {H}_{g}\in S_p(\mathcal {D}_v) if and only if \begin{aligned} \int _0^1 \left( (1-r)\int _{-\pi }^\pi |g'(r\text {e}^{i\theta })|^2\,\text {d}\theta \right) ^{\frac{p}{2}}\frac{{\text {d}}r}{1-r} <\infty . \end{aligned}

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