Remarks on generators of analytic semigroups

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL ^p , 1<p<∞, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MCS78-01245.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For L in some class of elliptic operators, we study weighted norm Lp inequalities for singular “non-integral” operators arising from L; those are the operators φ(L) for bounded holomorphic functions φ, the Riesz transforms ∇L−1/2 (or (−Δ)^1/2L−1/2) and its inverse L1/2(−Δ)^−1/2, some quadratic functionals gL and GL of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal Lp-regularity. For each, we obtain sharp or nearly sharp ranges of p using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.     

Generation of analytic semigroups in theL p topology by elliptic operators inR n

Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic semigroups of linear operators onL ^p(R ⁿ ), 1≦p≦∞. An explicit characterization of the domain is given for 1<p<∞. An application to parabolic problems is also included. This work has been partially supported by the Research Funds of the Ministero della Pubblica Istruzione. The authors are members of GNAFA (Consiglio Nazionale delle Ricerche).     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer     

On singular perturbation in non well posed problems

Summary Let L_εu and L ₀ v be the elliptic and “backward” heat operators defined by(1.1) and(1.2), respectively. The following question is considered for a pair of “non-well posed” initial-boundary value problems for L_ε and L ₀ : if u and v are the respective solutions, under what restrictions on the classes of admissible solutions and in what sense, if any, does u converge to v as ɛ →0? This research was supported in part by the National Science Foundation Grant No. GP 5882 with Cornell University.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain L ⁸ norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L ^q bounds, in the range 2⩽q⩽∞, for L ² - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp L ^q estimates in higher dimensions for a range of exponents q̅_n⩽q⩽∞. The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.     

Local a priori estimates in Lp for first order linear operators with nonsmooth coefficients

We prove local a priori estimates inL ^p , 1<p<∞, for first-order linear operators that satisfy the Nirenberg-Treves condition (p) and whose coefficients have Lipschitz continuous derivatives of order one. When the number of variables is two, only Lipschitz continuity of the coefficients is assumed. This extends toL ^p spaces estimates that were previously known forp=2. Examples show that the regularity required from the coefficients is essentially minimal. Research partially supported by CNPq.     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

Riesz transforms for Jacobi expansions

We define Riesz transforms and conjugate Poisson integrals associated with multi-dimensional Jacobi expansions. Under a slight restriction on the type parameters, we prove that these operators are bounded in L ^p , 1 < p < ∞, with constants independent of the dimension. Our tools are suitably defined g-functions and Littlewood-Paley-Stein theory, involving the Jacobi-Poisson semigroup and modifications of it. Research of both authors supported by the European Commission via the Research Training Network “Harmonic Analysis and Related Problems”, contract HPRN-CT-2001-00273-HARP. The first-named author was also supported by MNiSW Grant N201 054 32/4285.     

Decomposition of operator semigroups on W*-algebras

We consider semigroups of operators on a W^∗-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a “stable” and “reversible” part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W^∗-algebras.     

Interior a priori estimates in discreteL p norms for solutions of parabolic and elliptic difference equations

Summary We consider elliptic and parabolic difference operators and prove estimates in discrete L_p norms, 1<p<∞, which are analogues of known estimates for the corresponding differential operators. Let U be a solution in a bounded domain Ω of an elliptic or parabolic differential equation and let U_h be a solution of the discrete equation. Using the estimates, we prove under mild regularity assumptions that if U_h converges to U in some discrete L_p normp>1, then the difference quotients of U_h converge uniformly (on compact subsets of Ω) to the corresponding derivatives of U. Entrata in Redazione il 9 ottobre 1971.     

Large subspaces ofl ∞ n and estimates of the gordon-lewis constantand estimates of the gordon-lewis constant

Lower bounds are obtained for thegl constants and hence also for the unconditional basis constants of subspaces of finite dimensional Banach spaces. Sharp results are obtained for subspaces ofl _∞ ⁿ , while in the general case thegl constants of “random large” subspaces are related to the distance of “random large” subspaces to Euclidean spaces. In addition, a new isometric characterization ofl _∞ ⁿ is given, some new information is obtained concerningp-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern ^1/2. The research for this paper was begun while both authors were guests of the Mittag-Leffler Institute. Supported in part by NSF-MCS 79-03042.     

Sharp estimates for\bar \partial on convex domains of finite typeon convex domains of finite type

Let Ω be a bounded convex domain in C _n , with smooth boundary of finite typem. The equation is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toL ^p(Ω),p≥1. We solve the -equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted kernels are used; in order to reflect the geometry ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.     

Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations

We show that theL ^p norms, 0<p<∞, of the nontangenital maximal function and area integral of solutions and normalized adjoint solutions to second order nondivergence form elliptic equations, are comparable when integrated on the boundary of a Lipschitz domain with respect to measures, which are respectivelyA _∞ with respect to the corresponding harmonic measure or normalized harmonic measure. Both authors are supported by NSF     

Group invariance and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounded operators

The Hilbert and Riesz transforms can be characterized up to scalar as the translation invariant operators that satisfy additionally certain relative invariance of conformal transformation groups. In this article, we initiate a systematic study of translation invariant operators from group theoretic viewpoints, and formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar by means of relative invariance of affine subgroups. After providing some interesting examples of multiplier operators having “large symmetry”, we classify which of these examples can be extended to continuous operators on L ^p (R ⁿ ) (1 < p < ∞). T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science. A. Nilsson was partially supported by Japan Society for the Promotion of Science.     

Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup

We find necessary and sufficient conditions on a Banach spaceX in order for the vector-valued extensions of several operators associated to the Ornstein-Uhlenbeck semigroup to be of weak type (1, 1) or strong type (p, p) in the range 1<p<∞. In this setting, we consider the Riesz transforms and the Littlewood-Paleyg-functions. We also deal with vector-valued extensions of some maximal operators like the maximal operators of the Ornstein-Uhlenbeck and the corresponding Poisson semigroups and the maximal function with respect to the gaussian measure. In all cases, we show that the condition onX is the same as that required for the corresponding harmonic operator: UMD, Lusin cotype 2 and Hardy-Littlewood property. In doing so, we also find some new equivalences even for the harmonic case. The first and third authors were partially supported by CONICET (Argentina) and Convenio Universidad Autónoma de Madrid-Universidad Nacional del Litoral. The second author was partially supported by the European Commission via the TMR network “Harmonic Analysis”.     

An elliptic continuation result for harmonic functions in two dimensions

Let u be harmonic in a simply connected domainG ⊂ ℝ² and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth functionu ₁ and a second order uniformly elliptic operatorL with smooth coefficients in ℝ², satisfying:u ₁=u inK, Lu ₁=0 in ℝ². A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved.     

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”- and “DG”-norm formed by the “L ²(H ¹)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.     

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C _b (ℝ ^N ) and to evolution operators associated with nonautonomous second-order differential operators in C _b (ℝ ^N ) with time-periodic coefficients.     

Boundary estimates for elliptic systems with L<Superscript>1</Superscript>–data

We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions and L ¹–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge systems.