Fractional powers of Klein-Gordon-Fock operators in \(L_p\)-spaces
The authors are concerned with fractional powers of the Klein-Gordon-Fock operator \(\square+ m^2\), \(m>0\), in view of constructing an \(L^p\)-theory. More precisely, they consider the integral operator \[ (G^\alpha_m\varphi)(x)= a_{n,m}(\alpha) \int_{K^+_+} J_{{\alpha- n\over 2}}(mr(y)) r^{(\alpha- n)/2}(y)\times \varphi(x- y)dy,\quad \text{Re }\alpha> n-2, \] where \(a_{n,m}(\alpha)= m^{(n- \alpha)/2}\cdot 2^{(2- n-\alpha)/2}\pi^{1- n/2}\Gamma^{-1}\left({\alpha\over 2}\right)\), \(J_v(y)\) is a Bessel function of order \(v\) and \(K^+_+= \{y\in\mathbb{R}^n: r^2(y)\geq 0\), \(y_1\geq 0\}\) is the future light cone and using the notations \[ A= 4n(n- 1)/(2n^2+ (n+\text{Re }\alpha- 1)(n- 2)),\quad B= 4(n-1)/(3n+ \text{Re }\alpha- 5), \] \[ C= 2n(n-1)/(n^2+ \text{Re }\alpha(n- 2)),\quad D= 2(n- 1)/(\text{Re }\alpha+ n-2). \] They prove the following result: Theorem 1: Let \(n- 1\leq\text{Re }\alpha< n+1\) and \(A< p< B\). Then for \(\varphi(x)\in S\) the estimate \[ \| G^\alpha_m\varphi\|_q\leq C_{n,p}\|\varphi\|_p \] is valid with \(q= 2np/(2n- p(\text{Re }\alpha+ n-1))\). In the subintervals \(n-1\leq \text{Re }\alpha< (n^2+ n+ 2)/(n+ 2)\), \(A< p< D\) estimate (3) is valid for all \(q\) in the segment \([2np/(2n- p(\text{Re }\alpha+ n-1))\), \(np/(n- p\text{ Re }\alpha)]\). If \(n- 2< \text{Re }\alpha< n-1\) for \(n\leq 8\), and \(n(n- 5)/(n- 4)< \text{Re }\alpha< n- 1\) for \(n>8\), \(C< p< B\), then \[ \| G^\alpha_m\varphi\|_{L_r+ L_q}\leq C_{n,p}\|\varphi\|_p, \] \(\varphi(x)\in S\), \(r= np/(n- p\text{ Re }\alpha)\), \(q= 2np/(2n- p(\text{Re }\alpha+ n-1))\).
fractional powers of the Klein-Gordon-Fock operator
scientific article; zbMATH DE number 1050619