Estimate for the solution to the Cauchy problem for an ultrahyperbolic inequality (Q2477515)

The author considers the differential inequality \[ |Lu|^2\leq C(F^2+ u^2+ |\nabla_x u|^2), \] where \(x\in\Omega_1\times \Omega_2\equiv\Omega\) with the unknown function \(u(x)\). Here \(x= (x_1,\dots, x_n)\), and \(Lu= \sum^n_{i,j=1} (a_{ij}(x) u_{x_i})_{x_j}\), \(F= F(x)\) and the coefficients of the operator \(L\) satisfy the certain conditions. The author proves Carleman-type inequalities for \(L\) and additionally establishes a Hölder estimate for the corresponding Cauchy problem for \(L\).