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On mixed norm holomorphic grand and small spaces. (English) Zbl 1489.30051

The author deals with the mixed norm holomorphic grand and small Lebesgue spaces \(\mathcal A^{q;p),\theta}(\mathbb D)\) and \(\mathcal A^{q;(p,\theta}(\mathbb D)\) consisting in those holomporphic functions in the open unit disc \(\mathbb D\) belonging to the grand and small Lebesgue spaces \(L^{q;p),\theta}(\mathbb D)\) and \(L^{q;(p,\theta}(\mathbb D)\) which are defined as distributions \(f(r,e^{it})\) defined on \(\mathbb D\) such that the Fourier coefficients of the radial function \(f_n(r)\) belong to \(L^{p),\theta}((0,1))\) and \(L^{(p,\theta}(0,1))\) and the corresponding sequence \(\|f_n\|_{L^{p),\theta}((0,1))}\) or \(\|f_n\|_{L^{(p,\theta}((0,1))}\) belongs to \(\ell_q\) respectively. Recall here that \(L^{p),\theta}((0,1))\) is defined as the space of measurable functions \(\phi\) such \[\sup_{0<\varepsilon<p-1}\Big(\varepsilon^\theta\int_0^1 |\phi(r)|^{p-\varepsilon}dr\Big)^{1/(p-\varepsilon)}<\infty\] and \(L^{(p,\theta}((0,1))\) is the associate space. The main result of the paper shows that \(\|r^n\|_{L^{p),\theta}((0,1))}\approx n^{-1/p} \log^{-\theta/p}n\) and \(\|r^n\|_{L^{(p,\theta}((0,1))}\approx n^{-1/p} \log^{-\theta(1-1/p)}n\) as \(n\to \infty\). The author then applies this result to get a description of one equivalent norm in \(\mathcal A^{q;p),\theta}(\mathbb D)\) and \(\mathcal A^{q;(p,\theta}(\mathbb D)\) depending only on the size of the Taylor coefficients, and also another description of functions in such spaces using some generalized fractional integral of holomorphic functions with Taylor coefficients in \(\ell_q\). Finally an application to the Bergman projection is mentioned.

MSC:

30H20 Bergman spaces and Fock spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
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