On \(L^{p}\)-contractivity of Laguerre semigroups (Q384315)

Let \((\mathcal{X},\mu)\) be a \(\sigma\)-finite measure space and assume that \(\mathcal{D}\) is a linear subspace of the space \(\mathcal{M}\) of all measurable functions on \(\mathcal{X}\) such that it contains all \(L^p:=L^p(\mathcal{X},\mu)\) spaces, \(p\in[1,\infty]\). A symmetric diffusion semigroup is a family of linear operators \(\{T_t\}_{t\in[0,\infty)}\), \(T_0:=\text{Id}\), mapping jointly \(\mathcal{D}\) into \(\mathcal{M}\), and satisfying: (i) for each \(p\in[1,\infty]\), every \(T_t\) is a contraction on \(L^p\) and \(\{T_t\}_{t\in[0,\infty)}\) is a semigroup there; (ii) each \(T_t\) is a self-adjoint operator in \(L^2\); (iii) for each \(f\in L^2\), \(\lim_{\to 0^+}T_tf=f\) in \(L^2\).NEWLINENEWLINEA symmetric diffusion semigroup \(\{T_t\}_{t\in[0,\infty)}\) is called Markovian if it is positive and conservative, that is, satisfies additionally (iv) for each \(t\), \(T_tf\geq0\) if \(f\geq0\); (v) \(T_t1=1\) for all \(t\). Semigroups satisfying (i)--(iv) and (v') \(T_t1\leq1\) for all \(t\) replacing (v), are called submarkovian.NEWLINENEWLINESuppose that \(L\) is a positive self-adjoint operator on \(L^2\). The semigroup of operators NEWLINE\[NEWLINE\{\exp(-tL)\}_{t\in[0,\infty)},NEWLINE\]NEWLINE defined on \(L^2\) by means of the spectral theorem, is said to be an \(L^p\)-contractive semigroup if, for each \(p\in[1,\infty]\) and every \(t\in[0,\infty)\), one has NEWLINE\[NEWLINE\|\exp(-tL)f\|_{L^p}\leq\|f\|_{L^p},\;f\in L^2\cap L^p.NEWLINE\]NEWLINENEWLINENEWLINELet \(L^\alpha_{k}\) denote the Laguerre polynomial of degree \(k\in\mathbb{N}:=\{0,1,\dots\}\) and order \(\alpha\in(-1,\infty)\). The normalized Laguerre polynomial system, \(\{\hat{L}^\alpha_k: k\in\mathbb{N}\}\), is defined by NEWLINE\[NEWLINE\hat{L}^\alpha_k(x):=\left(\frac{\Gamma(k+1)} {\Gamma(k+\alpha+1)}\right)^{1\over 2}L^\alpha_{k}(x),\;x\in(0,\infty);NEWLINE\]NEWLINE the standard Laguerre function system, \(\{\mathcal{L}^\alpha_k: k\in\mathbb{N}\}\), is defined by NEWLINE\[NEWLINE\mathcal{L}^\alpha_k(x):=\hat{L}^\alpha_k(x)x^{\alpha/2}e^{-x/2},\;x\in(0,\infty).NEWLINE\]NEWLINENEWLINENEWLINELet \(\{\varphi^\alpha_k: k\in\mathbb{N}\}\) be the Laguerre function system of Hermite type, which is defined by \(\varphi^\alpha_k(x):=\sqrt{2}\hat{L}^\alpha_k(x^2)x^{\alpha+1/2}e^{-x^2/2}\), \(x\in(0,\infty)\). The corresponding multi-dimensional systems are formed by taking tensor products.NEWLINENEWLINELet \(\alpha:=(\alpha_1,\dots,\alpha_d)\in[0,\infty)^d\) for some positive integer \(d\) and \(\mathbb{R}_+:=(0,\infty)\). The differential operator related to the system \(\{\mathcal{L}^\alpha_k: k\in\mathbb{N}^d\}\) is defined by NEWLINE\[NEWLINEL^{\mathcal{L}}_\alpha:=-\sum_{i=1}^d\left(x_i\frac{\partial^2}{\partial x_i^2} +\frac{\partial}{\partial x_i}-\frac{x_i^2+\alpha_i^2}{4x_i}\right).NEWLINE\]NEWLINENEWLINEThe authors prove that \(\{T_t^{\alpha,\,\mathcal{L}}\}_{t\in[0,\infty)} :=\{\exp(-tL^{\mathcal{L}}_\alpha)\}_{t\in[0,\infty)}\) is an \(L^p(\mathbb{R}^d_+)\)-contractive submarkovian (but not Markovian) symmetric diffusion semigroup for all \(p\in[1,\infty]\). Precisely, for each \(p\in[1,\infty]\) and \(t\in(0,\infty)\), \(\|T_t^{\alpha,\,\mathcal{L}}\|_{L^p(\mathbb{R}^d_+)\to L^p(\mathbb{R}^d_+)} \leq(\cosh(t/2))^{-d}\). Moreover, let \(\alpha\in[-1/2,\infty)^d\). The differential operator related to the system \(\{\varphi^\alpha_k: k\in\mathbb{N}^d\}\) is defined by \(L^\varphi_\alpha:=-\Delta+|x|^2+\sum_{i=1}^d \frac1{x_i^2}(\alpha_i^2-\frac14)\). The authors show that, if \(\alpha\in(\{-1/2\}\cup[1/2,\infty))^d\), then \(\{T_t^{\alpha,\,\varphi}\}_{t\in[0,\infty)} :=\{\exp(-tL^{\varphi}_\alpha)\}_{t\in[0,\infty)}\) is an \(L^p(\mathbb{R}^d_+)\)-contractive submarkovian (but not Markovian) symmetric diffusion semigroup for all \(p\in[1,\infty]\) satisfying, for each \(p\in[1,\infty]\) and \(t\in(0,\infty)\), NEWLINE\[NEWLINE\|T_t^{\alpha,\,\varphi}\|_{L^p(\mathbb{R}^d_+)\to L^p(\mathbb{R}^d_+)} \leq(\cosh(2t))^{-d/2}.NEWLINE\]NEWLINENEWLINEIf \(\alpha_i\in(-1/2,1/2)\) for some \(i\in\{1,\dots,d\}\), then \(\{T_t^{\alpha,\,\varphi}\}_{t\in[0,\infty)}\) is not an \(L^p(\mathbb{R}^d_+)\)-contractive semigroup for all \(p\in[1,\infty]\), but there exists a constant \(c=c(\alpha)\in(1,\infty)\) such that the above estimate holds true with the right-hand side multiplied by \(c\).NEWLINENEWLINEThe authors also study several other Laguerre semigroups and find sharp ranges of the type parameter \(\alpha\) for which these semigroups are contractive on all \(L^p\) spaces (\(p\in[1,\infty]\)). In the end, the authors study a similar question for Bessel semigroups.