Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

Extensions of Well-Boundedness and <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-Scalarity

We consider some extensions of well-boundedness and C^m-scalarity by using fractional calculus, and prove some theorems accordingly. These results are applied to the usual Laplacian on Rⁿ and sub-Laplacians on nilpotent Lie groups. The research of first and second authors has been partly supported by the Project MTM2004-03036 of the M.C.YT.-DGI/F.E.D.E.R., Spain, and the Project E-12/25, D. G. Aragón, Spain. Part of the research of second author was developed in the Christian-Albrechts Universit?t in Kiel, while he was enjoying a HARP-postdoctoral position in the European Harmonic Analysis Network, HARP, IHP 2002-06.     

<Emphasis Type="Italic">K</Emphasis>th Roots of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators are Subscalar Operators of Order 4<Emphasis Type="Italic">k</Emphasis>

In this paper, we consider the special case of the question raised by Halmos (see below). In particular, we show that if T^k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T^k is p-hyponormal and σ(T) has nonempty interior in the plane, then T has a nontrivial invariant subspace.     

Maximal regularity for evolution equations in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem enjoys maximal regularity in weighted L _p -spaces with weights , where , if and only if it has the property of maximal L _p -regularity. Moreover, it is also shown that the derivation operator admits an -calculus in weighted L _p -spaces. Received: 26 February 2003     

A Note on <Emphasis Type="Italic">b</Emphasis>-Weakly Compact Operators

We consider a continuous operator T: E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Quasi Lp-Intersection Bodies

The purpose of this paper is to generalize the notion of intersection bodies to that of quasi Lp-intersection bodies. The Lp-analogs of the Busemann intersection inequality and the Brunn- Minkowski inequality for the quasi Lp-intersection bodies are obtained. The Aleksandrov Fenchel inequality for the mixed quasi Lp-intersection bodies is also established.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

Time-dependent diffusion operators on <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We study the Cauchy problem for time-dependent diffusion operators with singular coefficients on L¹-spaces induced by infinitesimal invariant measures. We give sufficient conditions on the coefficients such that the Cauchy-Problem is well-posed. We construct associated diffusion processes with the help of the theory of generalized Dirichlet forms. We apply our results in particular to construct a large class of Nelson-diffusions that could not been constructed before.     

An <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Theory of SPDEs on Lipschitz Domains

Stochastic partial differential equations are considered on Lipschitz domains. Existence and uniqueness results are given in weighted Sobolev spaces, and Hölder estimates of the solutions are also obtained. The number of derivatives of the solutions can be any real number, in particular, it can be negative and fractional. It is allowed that the coefficients of the equations blow up near the boundary.     

An Estimate of Growth Bound of Positive <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-Semigroup on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Space and its Applications

Let be a positive C ₀-semigroup on L ^p (Ω), with infinitesimal generator A. In this paper, it is proved that if there exists a such that and , where A ^* is the adjoint of A, then the growth bound of T(t) is upper bounded by b when p = 1, and by when 1 lt; p lt; α and c D(A), where . This is an operator version of a classical stability result on Z-matrix. As application examples, some new results on the asymptotic behaviours of population system and neutron transport system are obtained. Submitted: March 1, 2001?Revised: August 28, 2002     

Positive Solutions of Nonlinear Systems with <Emphasis Type="Italic">p</Emphasis>-Laplacian on Finite and Semi-infinite Intervals

Existence and localization results are established for systems of second-order differential equations with p -Laplacian on finite and semi-infinite intervals. For a compact interval, the compression and expansion conditions that are used are related to the first eigenvalue of the p-Laplacian.     

The Petty Projection Inequality for Lp-Mixed Projection Bodies

Recently, Lutwak, Yang and Zhang posed the notion of Lp-projection body and established the Lp-analog of the Petty projection inequality. In this paper, the notion of Lp-mixed projection body is introduced--the Lp-projection body being a special case. The Petty projection inequality, as well as Lutwak＇s quermassintegrals （Lp-mixed quermassintegrals） extension of the Petty projection inequality, is established for Lp-mixed projection body.     

Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an l_p-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on l_p that is quasinilpotent at a non-zero positive vector x₀ has a non-trivial invariant closed subspace. In this paper, it is proved that if is a collection of continuous operators with moduli on l_p that is finitely modulus-quasinilpotent at a non-zero positive vector x ₀ then and its right modulus sub-commutant have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on l _p whose moduli commute with a non-zero positive operator I on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace, so that all positive operators on l _p which commute with a non-zero positive operator S on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace. This research was supported by the Natural Science Foundation of Hunan Province of P. R. China (04JJ6004), the Foundation of Education Department of Hunan Province of P. R. China (04C002) and the Natural Science Foundation of P. R. China (10671147). Received: 4 December 2005 Revised: 19 June 2006     

<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer     

Lacunary Strong (<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">σ</Emphasis></Subscript><Emphasis Type="Italic">, p</Emphasis>)-Convergence

The definition of lacunary strongly convergence is extended to the definition of lacunary strong (A _σ , p)-convergence with respect to invariant mean when A is an infinite matrix and p = (p _i ) is a strictly positive sequence. We study some properties and inclusion relations.     

<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.