Some inequalities for continuous martingales associated with the Hermite polynomials (Q2730858)

Let \((X_t)_{t\geq 0}\) be a real-valued continuous martingale with quadratic variation process \(\langle X\rangle\). Consider the Hermite polynomials NEWLINE\[NEWLINEH_n (x,y) = (-y)^n \exp (x^2/2y)\frac{\partial^n}{\partial x^n} \exp (-x^2/2y),\quad n\geq 0.NEWLINE\]NEWLINE It is well-known that \((H_n(X_t,\langle X\rangle_t))_{t\geq 0}\) is a continuous local martingale for any \(n\geq 1\). For these martingales the author establishes weighted norm inequalities which generalize those given by \textit{E. Carlen} and \textit{P. Krée} [Ann. Probab. 19, No.~1, 354-368 (1991; Zbl 0721.60052)].