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A Weak-Type Inequality for Submartingales and Itô Processes

100%

Osękowski A.

tom Vol. 63, no. 1

73--88

Let α∈[0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate [WZÓR]. Here W is the weak-L∞space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Weighted weak-type inequality for martingales

100%

Osękowski A.

tom Vol. 65, no. 2

165--175

Let X = (Xt) t ≥ 0 be a bounded martingale and let Y = (Yt) t ≥ 0 be differentially subordinate to X. We prove that if 1 ≤ p < ∞ and W = (Wt) t ≥ 0 is an Ap weight of characteristic [W] Ap, then ∥Y∥Lp, ∞ (W) ≤ Cp [W]Ap∥X∥L∞(W). The linear dependence on [W]Ap is shown to be the best possible. The proof exploits a weighted exponential bound which is of independent interest. As an application, a related estimate for the Haar system is established.

Sharp norm inequalities for stochastic integrals in which the integrator is a nonnegative supermartingale

100%

Osękowski A.

tom Vol. 29, Fasc. 1

29--42

The paper is devoted to sharp inequalities between moments of nonnegative supermartingales and their strong subordinates. Analogous estimates hold true for stochastic integrals with respect to a nonnegative right-continuous supermartingale. Similar inequalities are established for smooth functions on Euclidean domains.

Weak-type inequality for the martingale square function and a related characterization of Hilbert spaces

100%

Osękowski A.

2011

tom Vol. 31, Fasc. 2

227--238

Let f be a martingale taking values in a Banach space B and let S(f) be its square function. We show that if B is a Hilbert space, then P(S(f) ≥1)≤√e∥f∥1and the constant √e is the best possible. This extends the result of Cox, who established this bound in the real case. Next, we show that this inequality characterizes Hilbert spaces in the following sense: if B is not a Hilbert space, then there is a martingale f for which the above weak-type estimate does not hold.

Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

100%

Osękowski A.

tom Vol. 61, no 3-4

209--218

Assume that u, v are conjugate harmonic functions on the unit disc of C, normalized so that u(0)=v(0)=0. Let u∗, |v|∗ stand for the one- and two-sided Brownian maxima of u and v, respectively. The paper contains the proof of the sharp weak-type estimate... [formula]. Actually, this estimate is shown to be true in the more general setting of differentially subordinate harmonic functions defined on Euclidean domains. The proof exploits a novel estimate for orthogonal martingales satisfying differential subordination.

Weak type inequality for the square function of a nonnegative submartingale

100%

Osękowski A.

tom Vol. 57, no 1

81-89

Let ƒ be a nonnegative submartingale and S(ƒ) denote its square function. We show that for any λ > 0, λP(S(ƒ) ≥ λ) ≤ π/2||ƒ||1, and the constant π/2 is the best possible. The inequality is strict provided ||ƒ||1 ≠ 0.

Moment Inequality for the Martingale Square Function

100%

Osękowski A.

2013

tom Vol. 61, no 2

169--180

Consider the sequence (Cn)n≥1 of positive numbers defined by C1=1 and Cn+1=1+C2n/4, n=1,2,…. Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound E|Mn|≤CnESn(M), n=1,2,…, and show that for each n, the constant Cn is the best possible.