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Hardy’s inequality and embeddings in holomorphic Triebel-Lizorkin spaces. (English) Zbl 0936.32004

The paper deals with holomorphic Triebel-Lizorkin spaces \(HF_s^{pq}\), \(0<p\), \(q\leq\infty\), \(s\in\mathbb{R}\), in the unit ball \(B\) of \(\mathbb{C}^n\). It is shown that \[ \sum_{n\geq 0}|a_n|/(n+1) \lesssim\|\sum_{n\geq 0}a_nz^n \|_{HF_0^{1\infty}}, \] which improves the classical Hardy’s inequality for holomorphic functions in the Hardy space \(H^1=H F_0^{1 2}\) in the disc. As a consequence of this result, it is obtained that if \(f\in HF_0^{1\infty} (B)\) and \(f=\sum_kf_k\) is the homogeneous expansion of \(f\) at 0, then \(\sum_{k\geq 0}\|f_k\|_{L^1(S)}/(k+1)\lesssim\|f\|_{H F_0^{1 \infty}}\).
Moreover, it is proved that \((HF_s^{1q})^* =HF_s^{\infty q'}\), \(1 \leq q<\infty\), where the pairing is given by \[ (f,g)= \int_B(I+R)^k f(I+ \overline R)^k\overline g\bigl(1-|z|\bigr)^{2(k-s)-1}dV(z),\;k>s \] \((I\) denotes the identity operator and \(R\) the radial derivative). This result includes the classical duality theorem \((H^1)^*=BMOA=HF_0^{\infty 2}\).
Finally, it is given some embeddings between holomorphic Triebel-Lizorkin and Besov spaces, which are used to obtain some trace results.

MSC:

32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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