Embedding Bergman spaces into tent spaces

By J. A. Pel'aez, Jouni Rattya, Kian Sierra. 2015

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Let $$A^p_\omega $$Aωp denote the Bergman space in the unit disc $$\mathbb {D}$$D of the complex plane induced by a radial weight $$\omega $$ω with the doubling property $$\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds$$∫r1ω(s)ds≤C∫1+r21ω(s)ds. The tent space $$T^q_s(\nu ,\omega )$$Tsq(ν,ω) consists of functions such that $$\begin{aligned} \begin{aligned} \Vert f\Vert _{T^q_s(\nu ,\omega )}^q =\int _\mathbb {D}\left( \int _{\varGamma (\zeta )}|f(z)|^s\,d\nu (z)\right) ^\frac{q}{s}\omega (\zeta )\,dA(\zeta ) <\infty ,\quad 0<q, \; s<\infty . \end{aligned} \end{aligned}$$‖f‖Tsq(ν,ω)q=∫D∫Γ(ζ)|f(z)|sdν(z)qsω(ζ)dA(ζ)<∞,0<q,s<∞.Here $$\varGamma (\zeta )$$Γ(ζ) is a non-tangential approach region with vertex $$\zeta $$ζ in the punctured unit disc $$\mathbb {D}{\setminus }\{0\}$$D\{0}. We characterize the positive Borel measures $$\nu $$ν such that $$A^p_\omega $$Aωp is embedded into the tent space $$T^q_s(\nu ,\omega )$$Tsq(ν,ω), where $$1+\frac{s}{p}-\frac{s}{q}>0$$1+sp-sq>0, by considering a generalized area operator. The results are provided in terms of Carleson measures for $$A^p_\omega $$Aωp.

Subfields: Applied Mathematics

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