The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity

In this paper we establish a general weighted L ^q -theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L ^q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L ^p -regularity of in the respective weighted L ^q -spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L ^p -regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L _p -regularity. Preprint (1999)] with the fact that for L ^q -spaces -boundedness is implied by weighted estimates.     

Strictly singular inclusions into $$L^1 + L^\infty$$

It is given a complete characterization of the strict singularity and the disjoint strict singularity of the inclusions E ↪ L ¹ + L ^∞ for the class of rearrangement invariant function spaces E on the interval. Their relationship is also analyzed. Suitable criteria are given involving the scale of order continuous weak L ^p -spaces for . Evgeny M. Semenov was partially supported by RFBR (Russia), grant 05-01-00629, by Universities of Russia grant 04.01.051 and by a Spanish Ministry of Education grant SAB2002.     

On weighted L p -spaces of vector-valued entire analytic functions

The weighted L _p -spaces of entire analytic functions are generalized to the vector-valued setting. In particular, it is shown that the dual of the space is isomorphic to when the function χ _K is an L _p,ρ (E)-Fourier multiplier. This result allows us to give some new characterizations of the so-called UMD-property and to represent several ultradistribution spaces by means of spaces of vector sequences. J. Motos is partially supported by DGI (Spain), Grant BFM 2002-04013 and Grant MTEM 2005-08350-C03-03.     

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier–Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, . For the Navier–Stokes equations the convergence of the velocity to the self-similar singularity is in L ^q (B(z,r)) for some , where the ball of radius r is shrinking toward a possible singularity point z at the order of as t approaches to T. In the convergence case with we present a simple alternative proof of the similar result in Hou and Li in arXiv-preprint, math.AP/0603126. This work was supported partially by KRF Grant(MOEHRD, Basic Research Promotion Fund) and the KOSEF Grant no. R01-2005-000-10077-0.     

Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data

In this paper we investigate vector-valued parabolic initial boundary value problems , subject to general boundary conditions in domains G in with compact C ^2m -boundary. The top-order coefficients of are assumed to be continuous. We characterize optimal L ^p -L ^q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on and the Lopatinskii–Shapiro condition on are necessary for these L ^p -L ^q -estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.      

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

Dominated Compactness Theorem in Banach Function Spaces and its Applications

A famous dominated compactness theorem due to Krasnosel’skiĭ states that compactness of a regular linear integral operator in L ^p follows from that of a majorant operator. This theorem is extended to the case of the spaces , with variable exponent p(·), where we also admit power type weights . This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces is applied to fractional integral operators over bounded open sets. Submitted: June 6, 2007. Accepted: November 20, 2007.     

A Unified Fixed Point Theory in Generalized Convex Spaces

Let B be the class of ＇better＇ admissible multimaps due to the author. We introduce new concepts of admissibility （in the sense of Klee） and of Klee approximability for subsets of G-convex uniform spaces and show that any compact closed multimap in B from a G-convex space into itself with the Klee approximable range has a fixed point. This new theorem contains a large number of known results on topological vector spaces or on various subclasses of the class of admissible G-convex spaces. Such subclasses are those of O-spaces, sets of the Zima-Hadzic type, locally G-convex spaces, and LG-spaces. Mutual relations among those subclasses and some related results are added.     

Pointwise bounds for L 2 eigenfunctions on locally symmetric spaces

We prove pointwise bounds for L ² eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with -rank one if the corresponding eigenvalues lie below the continuous part of the L ² spectrum. Furthermore, we use these bounds in order to obtain some results concerning the L ^p spectrum.     

Maximal L p –L q Regularity for Parabolic Partial Differential Equations on Manifolds with Cylindrical Ends

We give a short, simple proof of maximal L^p–L^q regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of -boundedness for the resolvent.      

<Emphasis Type="Italic">L</Emphasis>-functions of symmetric products of the Kloosterman sheaf over Z

The classical n-variable Kloosterman sums over the finite field F _p give rise to a lisse -sheaf Kl _n+1 on , which we call the Kloosterman sheaf. Let L _p (G _{m, F p} , Sym ^k Kl _n+1, s) be the L-function of the k-fold symmetric product of Kl _n+1. We construct an explicit virtual scheme X of finite type over Spec Z such that the p-Euler factor of the zeta function of X coincides with L _p (G _{m, F p} , Sym ^k Kl _n+1, s). We also prove similar results for and . The research of L. Fu is supported by the NSFC (10525107).     

Multipliers in Besov Spaces and in LpΩ-Spaces (The Cases 0<p≦1 and p=∞)

The paper contains sufficient conditions for multipliers in two types of quasi-BANACH spaces: (i) weighted L_p-spaces of entire analytic functions of exponential type; 0<p≦∞, (ii) Besov spaces B, where ?∞<s<∞; 0<p≦∞; 0<q≦∞.     

Local well-posedness for the homogeneous Euler equations

We introduce Triebel-Lizorkin-Lorentz function spaces, based on the Lorentz L^p,q-spaces instead of the standard L^p-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rⁿ,n≥2. As a corollary we obtain global existence of solutions to the 2D Euler equations in the Triebel-Lizorkin-Lorentz space. For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J.M. Bony.     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Strong density results in trace spaces of maps between manifolds

We deal with strong density results of smooth maps between two manifolds and in the fractional spaces given by the traces of Sobolev maps in W ^1,p .     

The multi-dimensional pencil phenomenon for Laguerre heat-diffusion maximal operators

We investigate in detail the mapping properties of the maximal operator associated with the heat-diffusion semigroup corresponding to expansions with respect to multi-dimensional standard Laguerre functions . Our interest is focused on the situation when at least one coordinate of the type multi-index α is smaller than 0. For such parameters α the Laguerre semigroup does not satisfy the general theory of semigroups, and the behavior of the associated maximal operator on L ^p spaces is found to depend strongly on both α and the dimension. A. Nowak was supported in part by MNiSW Grant N201 054 32/4285.     

A Class of Operators on Weighted Bergman Spaces

This paper studies the connection between an analytic function φ on the unit disk and a certain type of operators induced by φ on weighted Bergman spaces A ^α of analytic functions on . The general theme is the interplay between smoothness conditions on φ and the S _p -norms of the operators induced by φ.      

Unconditionality in spaces of smooth functions

Our main result shows that subspaces of L¹([0, 1]) on which the blow-up operators act compactly are isometric to dual spaces, and their natural predual belongs to the Banach-Mazur closure of quotient spaces of . Related general results are shown for subspaces X of or of reflexive K?the function spaces, which imply that when X consists of smooth functions it embeds into a Banach space with an unconditional basis. Received: 25 September 2008     

On a class of degenerate elliptic operators in <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> spaces with respect to invariant measures

We consider a class of second order degenerate elliptic operators arising from second order stochastic differential equations in perturbed by noise. We study realizations of such operators in L ¹ spaces with respect to their (explicit) invariant measure, proving that they are m-dissipative.      

H ∞-calculus for the Stokes operator on unbounded domains

We consider the Stokes operator A on unbounded domains of uniform C ^1,1-type. Recently, it has been shown by Farwig, Kozono and Sohr that – A generates an analytic semigroup in the spaces , 1 < q < ∞, where for q ≥ 2 and for q ∈ (1, 2). Moreover, it was shown that A has maximal L ^p -regularity in these spaces for p ∈ (1,∞). In this paper we show that ɛ + A has a bounded H ^∞-calculus in for all q ∈ (1, ∞) and ɛ > 0. This allows to identify domains of fractional powers of the Stokes operator. Received: 12 October 2007