In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John-Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.
The obtained periodic magnetic-field dependencies of the critical current Ic+(Φ/Φ0), Ic−(Φ/Φ0), measured in opposite directions on asymmetric superconducting aluminum rings, allow to explain observed earlier quantum oscillations of a dc voltage as a result of alternating current rectification. It is found, that the high efficiency of the rectification of both individual rings and ring systems is connected to a hysteresis of the current-voltage characteristics. The asymmetry of the currentvoltage characteristics providing the rectification effect is due to the relative shifts of the magnetic dependencies Ic−(Φ/Φ0) = Ic+(Φ/Φ0 + ∆φ) of the critical current measured in opposite directions. This shift means that position of Ic+(Φ/Φ0) and Ic−(Φ/Φ0) minimums does not correspond to n+0.5 magnetic flux quantum Φ0 which is in the direct contradiction with measurement results of the Little-Parks resistance oscillations. Despite of this contradiction the amplitude of the critical current anisotropy oscillations Ic,an(Φ/Φ0) = Ic+(Φ/Φ0) − Ic−(Φ/Φ0) and its variations with temperature correspond to expected amplitude of the persistent current oscillations and to its variations with temperature.
In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John-Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.
Rubio de Francia proved the one-sided Littlewood-Paley inequality for arbitrary intervals in L p , 2 ≤ p < ∞. In this article, such an inequality is proved for the Walsh system. Formulation of the resultFirst, we make some agreement about our notation. From now on, by the space L p we mean the space L p ([0, 1]). Also, by L p (l 2 ) we mean L p ([0, 1], l 2 ) (i.e., the space of l 2 -valued functions on the interval [0, 1]).Let I m be mutually disjoint intervals in Z (here and below, we assume that m runs over some finite or countable set). In 1983, Rubio de Francia proved (see [1]) thatwhere the constant C p does not depend on the intervals I m or the function f . It is worth noting that he considered the whole line R rather than the interval [0, 1] (so I m were intervals in R, not in Z), but this fact did not play a significant role in his considerations. By duality, estimate (1) is equivalent to the following:(where {f m } is a sequence of functions such that supp f m ⊂ I m . In fact, it is already known that estimate (2) remains true for p ∈ (0, 1] (see [2] for p = 1 and [3] for all p ∈ (0, 1]). Our goal is to prove an analogue of (2) for the situation where we use the Walsh system instead of the exponential functions. We give the corresponding definition.Definition 1. The Walsh system {w n } n∈Z+ consists of step functions on the interval [0, 1] that are defined as follows. First, we set w 0 ≡ 1. Next, for any index n > 0 we consider its dyadic decomposition n = 2 k1 + · · · + 2 ks , k 1 > k 2 > · · · > k s ≥ 0, and setKey words and phrases. Calderón-Zygmund operator, martingales. This work was carried out during the tenure of an ERCIM "Alain Bensoussan" Fellowship Programme.
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