Bergman spaces of temperature functions on a cylinder (Q1811899)

This article, being a part of the author's doctoral thesis at UNAM, deals with weighted Bergman-type spaces \(b_\beta^p (S_T)\) consisting of temperature functions on the cylinder \(S_T= S^1\times (0,T)\) and belonging to \(L^p (\Omega_T, t^\beta\,dx\,dt)\), where \(\Omega_T= (0,2)\times (0,T)\). Since these spaces are shown to be Banach spaces, the Bergman projection and the reproducing kernel exist in this setting. In order to prove boundedness of (an analogy of) the Bergman projection from \(L^p (\Omega_T)\) onto \(b^p (S_T)\), a family of reproducing kernels and certain integral operators \(P_\alpha\) are constructed which turn out to be continuous projections form \(L^p (\Omega_T, t^\beta\,dx\,dt)\) onto \(b_\beta^p (S_T)\), for \(\alpha,\beta> -1\) and \(\max [1, (1+\beta)/ (1+\alpha)]< p< \infty\). The proof of the boundedness of these projections is based on a variant of Schur's lemma providing a sufficient condition for the boundedness of integral operators defined on \(L^p (\Omega,d\mu)\) \((1< p<\infty\), \(\mu\) being a \(\sigma\)-finite measure), and on the theory of Fourier multipliers. Furthermore, by utilization of the results on the \(P_\alpha\)'s, the duality \(b_\beta^p (S_T)^*= b_{\beta'}^{p'} (S_T)\) is obtained where \(\alpha\), \(\beta\), \(p\) are as above, and \(1/p+ 1/p'= 1\), \(\beta'= (\alpha- \beta/p)p'\).