Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript> convergence for <Emphasis Type="Italic">B</Emphasis>-valued random elements

The paper investigates L ^p convergence and Marcinkiewicz-Zygmund strong laws of large numbers for random elements in a Banach space under the condition that the Banach space is of Rademacher type p, 1 < p < 2. The paper also discusses L ^r convergence and L ^r bound for random elements without any geometric restriction condition on the Banach space.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-boundedness of pseudo-differential operators with non-regular symbols

In this paper, we consider the continuity property of pseudo-differential operators with symbols whose Fourier transforms have compact support. As applications, we obtain the L ^p -boundedness for symbols in Besov spaces and in modulation spaces.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

Mixed norm estimate for Radon transform on weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We will discuss about the mapping property of Radon transform on L ^p spaces with power weight. It will be shown that the Pitt’s inequality together with the weighted version of Hardy-Littlewood-Sobolev lemma imply weighted inequality for the Radon transform.     

Sobolev Inequalities on Forms and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript>-Cohomology on Complete Riemannian Manifolds

We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable geometric conditions. Moreover, we establish some L ^p,q -estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing theorems of the L ^p,q -cohomology and prove the L ^q -solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions.     

Summability Kernels for <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Multipliers

In this article we study the problem of extending Fourier Multipliers on L ^p (T) to those on L ^p (R) by taking convolution with a kernel, called a summability kernel. We characterize the space of such kernels for the cases p = 1 and p = 2. For other values of p we give a necessary condition for a function to be a summability kernel. For the case p = 1, we present properties of measures which are transferred from M(T) to M(R) by summability kernels. Furthermore it is shown that every l _p sequence can be extended to some L ^q (R) multipliers for certain values of p and q.     

On convex risk measures on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L ^∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on L ^p spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on L ^p we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for L ^p risks.     

SOME RESULTS ABOUT BEST COAPPROXIMATION IN L^P（S,X）

As a counterpart to best approximation in normed linear spaces, the best coapproximation was introduced by Franchetti and Furi. In this paper, we shall consider the relation between coproximinality M in X and L ^p (S,M) in L ^p (S,X). Finally we give some results in cochebyshev subspaces and additional subspaces.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted L ^p -space L ^p (G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for L ^p (G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and L ^p (G, ω) is a Banach algebra, then G is σ-compact.     

Lambert multipliers between <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

In this paper Lambert multipliers acting between L ^p spaces are characterized by using some properties of conditional expectation operator. Also, Fredholmness of corresponding bounded operators is investigated.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

Direct approach to <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates in homogenization theory

We derive interior L ^p -estimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our approach relies on BMO-estimates and an interpolation argument, gradients are treated with the help of finite differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an L ^p -corrector result for approximate solutions.      

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-convergence of Lagrange interpolation on the semiaxis

In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence in weighted L ^p -spaces.      

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> integrability of trigonometric series with special varying coefficients

Very recently, Yu, Le and Zhou introduced the so called △B1^＊ and △B2^＊ conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the classical results of Chen on the necessary and sufficient conditions of the Lp integrability of trigonometric series. In fact, the present paper gives the first result on the necessary and sufficient conditions of the Lp integrability of trigonometric series, where coefficients may have different signs.     

A Noncommutative Version of <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and Characterizations of Subdiagonal Algebras

Let M{\mathcal M} be a σ-finite von Neumann algebra and \mathfrak A{\mathfrak A} a maximal subdiagonal algebra of M{\mathcal M} with respect to a faithful normal conditional expectation F{\Phi} . Based on Haagerup’s noncommutative L ^p space L^p(M){L^p(\mathcal M)} associated with M{\mathcal M} , we give a noncommutative version of H ^p space relative to \mathfrak A{\mathfrak A} . If h ₀ is the image of a faithful normal state j{\varphi} in L¹(M){L^1(\mathcal M)} such that j°F = j{\varphi\circ \Phi=\varphi} , then it is shown that the closure of {\mathfrak Ah₀^\frac1p}{\{\mathfrak Ah_0^{\frac1p}\}} in L^p(M){L^p(\mathcal M)} for 1 ≤ p < ∞ is independent of the choice of the state preserving F{\Phi} . Moreover, several characterizations for a subalgebra of the von Neumann algebra M{\mathcal M} to be a maximal subdiagonal algebra are given.     

Essential norm of substitution operators on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

In this paper we determine the lower and upper estimates for the essential norm of finite sum of weighted Frobenius-Perron and weighted composition operators on L ^p spaces under certain conditions.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.