The $p$-Weil-Petersson Teichm\"uller space and the quasiconformal extension of curves

We consider the correspondence between the space of $p$-Weil-Petersson curves $\gamma$ on the plane and the $p$-Besov space of $u=\log {\gamma }^{\mathrm{\prime }}$ on the real line for $p>1$. We prove that the variant of the Beurling-Ahlfors extension defined by using the heat kernel yields a holomorphic map for $u$ on a domain of the $p$-Besov space to the space of $p$-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichm"uller projection from the space of $p$-integrable Beltrami coefficients to the $p$-Weil-Petersson Teichm"uller space.