Operators associated with the Jacobi semigroup
This paper focusses on the study of potential operators, fractional integrals, fractional derivatives and Carleson measures in the Jacobi setting. For \(d\in\mathbb N\), \(x\in(-1,1)^d\) and \(\alpha,\beta\in (-1,\infty)^d\), let \(\mu_d^{\alpha,\beta}(dx)\equiv\prod_{i=1}^d (1-x_i)^{\alpha_i} (1+x_i)^{\beta_i}\,dx_i\) be the Jacobi measure on \(\mathbb R^d\), \[ {\mathcal L}^{\alpha,\beta}\equiv \sum_{i=1}^d \left[(1-x_i^2) \partial^2_{x_i}+(\beta_i-\alpha_i- (\alpha_i+\beta_i+2)x_i)\partial_{x_i}\right] \] the Jacobi differential operator, and \(\{p^{\alpha,\beta}_v\}_v\) the set of all normalized Jacobi polynomials. Denote by \(C^{\alpha,\beta}_n\) the closed subspaces of \(L^2(\mu_d^{\alpha,\beta})\) generated by \(\{p^{\alpha,\beta}_v: |v|=n\}\) and let \(J^{\alpha,\beta}_n\) be the orthogonal projection from \(L^2(\mu_d^{\alpha,\beta})\) onto \(C^{\alpha,\beta}_n\). The author first proves that \(J^{\alpha,\beta}_n\) is bounded on \(L^p(\mu_d^{\alpha,\beta})\) for all \(1<p<\infty\). Via this, the author establishes the \(L^p(\mu_d^{\alpha,\beta})\)-boundedness of potential operators \(U^{\alpha,\beta}_k\) and \(U^{s,\alpha,\beta}_k\), defined, respectively, by \[ U^{\alpha,\beta}_k f\equiv \int_0^\infty \exp (t{\mathcal L}^{\alpha,\beta})(I-\cdots-J^{\alpha,\beta}_{k-1})f\,dt \] and \[ U^{s,\alpha,\beta}_k\equiv \int_0^\infty \int_0^\infty \exp (u{\mathcal L}^{\alpha,\beta})(I-\cdots-J^{\alpha,\beta}_{k-1})f \eta^s_t(du)\,dt \] for all polynomials \(f\), where \(\eta^s_t\) is a Borel measure on \([0,\infty)\) satisfying \(\int_0^\infty e^{-wu} \eta^s_t(du)=e^{-w^s t}\), and then extends the celebrated Meyer's multiplier theorem to the Jacobi expansions. As applications of these conclusions, the author introduces the fractional integral, the fractional derivative \(D^{\mu^{\alpha,\beta}}_s\) of order \(s>0\) and the potential space \(L^p_s(\mu_d^{\alpha,\beta})\) associated to Jacobi expansions, and shows that \(f\in L^p_s(\mu_d^{\alpha,\beta})\) if and only if \(D^{\mu^{\alpha,\beta}}_s f\in L^p(\mu_d^{\alpha,\beta})\). The author also establishes a version of Calderón's reproduction formula via the Jacobi semigroup \(\{T^{\alpha,\beta}_t\}_{t\geq0}\equiv \{\exp(t{\mathcal L}^{\alpha,\beta})\}_{t\geq0}\) and then introduces the Jacobi-Besov spaces \({\dot B}^{s,p}_{q}(\mu_d^{\alpha,\beta})\) and the Jacobi-Triebel-Lizorkin spaces \({\dot F}^{s,p}_{q}(\mu_d^{\alpha,\beta})\) in the Jacobi setting. The Lebesgue spaces \(L^p(\mu_d^{\alpha,\beta})\) and the potential spaces \(L^p_s(\mu_d^{\alpha,\beta})\) are proved to be particular cases of Jacobi-Triebel-Lizorkin spaces. Finally, the author establishes the Carleson measure characterization for the space \({\dot F}^{0,2}_{\infty}(\mu_d^{\alpha,\beta})\).
scientific article; zbMATH DE number 5639051
scientific article