In this chapter we shall first study two basic results in interpolation of operators in Lp spaces, the Riesz-Thorin theorem and the Marcinkiewicz interpolation theorem (diagonal case). As a ...consequence of the former we shall prove the Hardy-Littlewood-Sobolev theorem for Riesz potentials. In this regard we need to introduce one of the fundamental tools in harmonic analysis, the Hardy-Littlewood maximal function. In Section 2.4 we shall prove the Mihlin multiplier theorem.
Some real analysis Giaquinta, Mariano; Martinazzi, Luca
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
Book Chapter
We collect in this chapter some facts of real analysis that will be relevant for us in the sequel.
A driving system is presented for generating a uniform luminance picture on passive matrix organic light emitting diode (PMOLED) panels in this paper. Especially, the controller overcomes the problem ...of luminance non-uniformity on displaying pictures. Because the controller is a voltage type driver, the output impedance of the driver is much less than that of the current-type driver. Hence, the controller provides a better electron-optical response than those of traditional current drivers. According to the linear interpolation theorem, we designed an area compensated look up table (ACLUT) with compensation data to improve the luminance uniformity on lighting a PMOLED panel. By applying the proposed driver to light PMOLED panels, the luminance uniformity of the PMOLED panel is improved from 88% to 98%
In this paper, a single-antecedent/succedent sequent calculus NL for first-order Nelsonian paraconsistent quantum logic is investigated. The logic under consideration is regarded as a combination of ...both Nelson's paraconsistent four-valued logic and Dalla Chiara and Giuntini's paraconsistent quantum logic. The duality and cut-elimination theorems for NL are proved. Decidability, some constructive properties, some constructible falsity properties, and Craig interpolation property are shown for NL. An extend NL with some naive comprehension rules in the naive set theory is also investigated.
A natural deduction formulation is given for the intermediate logic called MH by Gabbay in 4. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest ...intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.
A Banach space $Z$ has the interpolation property with respect to the pair $(X, Y)$ if each $T$, which is a bounded linear operator from $X$ to $X$ and from $Y$ to $Y$, can be extended to a bounded ...linear operator from $Z$ to $Z$. If $X = L^p, Y = L^\infty$ we give a necessary and sufficient condition for a Banach function space $Z$ on $(0, l), 0 < l \leqq + \infty$, to have this property. The condition is that $g \prec^pf$ and $f \in Z$ should imply $g \in Z$; here $g \prec^pf$ means that $g^{\ast p} \prec f^{\ast p}$ in the Hardy-Littlewood-Polya sense, while $h^\ast$ denotes the decreasing rearrangement of the function $|h|$. If the norms $\|T\|_X, \|T\|_Y$ are given, we can estimate $\|T\|_Z$. However, there is a gap between the necessary and the sufficient conditions, consisting of an unknown factor not exceeding $\lambda_p, \lambda_p \leqq 2^{1/q}, 1/p + 1/q = 1$. Similar results hold if $X = L^1, Y = L^q$. For all these theorems, the complete continuity of $T$ on $Z$ is assured if $T$ has this property on $X$ or on $Y$, and if $Z$ satisfies a certain additional necessary and sufficient condition, expressed in terms of $\|\sigma_a\|_z, a > 0$, where $\sigma_a$ is the compression operator $\sigma_af(t) = f(at), 0 \leqq t < l$.
An alternative is provided to a recently published method of Benjauthrit and Reed for calculating the coefficients of the polynomial expansion of a given function. The method is exhibited for ...functions of one and two variables. The relative advantages and disadvantages of the two methods are discussed. Some empirical results are given for GF(9) and GF(16). It is shown that functions with DON'T CARE states are represented by a polynomial of minimal degree by this method.
We give here some of the basic properties of the classes$\{\Phi_r\}, \{\Psi_r\}, - 1 < r < 1$, of dilation operators acting in rearrangement-invariant spaces X on the circle. It is shown that to each ...space X there correspond two numbers ξ, η, called indices, which satisfy 0 ≤ η ≤ ξ ≤ 1; these numbers represent the rate of growth or decay of | Ψr| as r → ± 1. By using the operators Ψrto obtain estimates for certain averaging operators Aγ, we are able to show that the indices (ξ, η) coincide with the Boyd indices (α, β). As a consequence, we obtain a Marcinkiewicz-type interpolation theorem for rearrangement-invariant spaces on the circle.
Amalgamation of Polyadic Algebras Johnson, James S.
Transactions of the American Mathematical Society,
01/1970, Letnik:
149, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The main result of the paper is that for $I$ an infinite set, the class of polyadic $I$-algebras (with equality) has the strong amalgamation property; i.e., if two polyadic $I$-algebras have a given ...common subalgebra they can be embedded in another algebra in such a way that the intersection of the images of the two algebras is the given common subalgebra.
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