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XIII: 26, 294-306, LNM 721 (1979)

**BONAMI, Aline**; **LÉPINGLE, Dominique**

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $**P**$ two functionals of a $**P**$-martingale---remains true, possibly with new constants, when $**P**$ is replaced by an equivalent law $Z.**P**$. To this order, the ``weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities, 1326, 1327, 1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIII: 27, 307-312, LNM 721 (1979)

**IZUMISAWA, Masataka**; **SEKIGUCHI, Takesi**

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in*Tôhoku Math. J.* **29**, 1977. For a modern reference see also Kazamaki, *Continuous Exponential Martingales and $\,BMO$,* LNM. 1579, 1994

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIII: 28, 313-331, LNM 721 (1979)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

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XV: 20, 285-289, LNM 850 (1981)

**CHOU, Ching Sung**

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

Retrieve article from Numdam

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIII: 27, 307-312, LNM 721 (1979)

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIII: 28, 313-331, LNM 721 (1979)

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

Retrieve article from Numdam

XV: 20, 285-289, LNM 850 (1981)

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

Retrieve article from Numdam