Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {ℝ}^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(ℝ;L²\left(\Sigma \right))$, q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

In this paper we deal with the stationary Navier-Stokes problem in a domain $\Omega$ with compact Lipschitz boundary $\partial \Omega$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial \Omega$, with possible countable exceptional set, provided $a$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $\Omega$ bounded.

We prove the existence of solutions to the evolutionary Stokes system in a bounded domain Ω ⊂ ℝ³. The main result shows that the velocity belongs either to $W_p^{2s+2,s+1}\left({\Omega }^T\right)$ or to $B_{p,q}^{2s+2,s+1}\left({\Omega }^T\right)$ with p > 3 and s ∈ ℝ₊ ∪ 0. The proof is divided into two steps. First the existence in $W_p^{2k+2,k+1}$ for k ∈ ℕ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces...

We consider sequences of solutions of the Navier-Stokes equations in ${ℝ}^3$, associated with sequences of initial data bounded in ${\dot{H}}^{1/2}$. We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in ${\dot{H}}^{1/2}$, up to a remainder term small in $L^3$; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If $𝒜$ is an “admissible” space (in...

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f\in L_{2,-\mu }$, $v₀\in H_{-\mu }¹$, μ ∈ (0,1). We prove an estimate of the velocity in the $H_{-\mu }^{2,1}$ norm and of the gradient of the pressure in the norm of $L_{2,-\mu }$. We apply the Fourier transform with respect to the variable along...

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$, namely, it is represented by a stress tensor $T(Du,p):=\nu (p,|D{|}^2)D$ which satisfies $r$-growth condition with $r\in (1,2]$. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for...

We study embeddings of spaces of Besov-Morrey type, $i{d}_{\Omega }{:}_{p₁,u₁,q₁}^{s₁}\left(\Omega \right){\hookrightarrow }_{p₂,u₂,q₂}^{s₂}\left(\Omega \right)$, where $\Omega \subset {ℝ}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $i{d}_{\Omega }$. This continues our earlier studies relating to the case of ${ℝ}^d$. Moreover, we also characterise embeddings into the scale of $L_p$ spaces or into the space of bounded continuous functions.

CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into $W^{m,p}\left(\Omega \right)$ into $L^S\left(\Omega \right)$ (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding $W^{m,p}\left(\Omega \right)$ into $L^{\phi }\left(\Omega \right)$...............................................................

Let $\Theta :={\theta }_I^e:e\in E,I\in D$ be a decomposition system for $L₂\left({ℝ}^d\right)$ indexed over D, the set of dyadic cubes in ${ℝ}^d$, and a finite set E, and let $\Theta ̃:=\Theta {̃}_I^e:e\in E,I\in D$ be the corresponding dual functionals. That is, for every $f\in L₂\left({ℝ}^d\right)$, $f={\sum }_{e\in E}{\sum }_{I\in D}⟨f,\Theta {̃}_I^e⟩{\theta }_I^e$. We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨f,\Theta {̃}_I^e⟩$, e ∈ E, I ∈ D. Typical examples of such decomposition...

A two-sided sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m}+c_{n-m}=2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ is bounded if $su{p}_{n\in \mathbb{Z}}\left|\right|cₙ\left|\right|<\infty$. A (bounded) group decomposition for a cosine sequence $c={\left(cₙ\right)}_{n\in \mathbb{Z}}$ is a representation of c as $cₙ=(bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $su{p}_{n\in \mathbb{Z}}\left|\right|bⁿ\left|\right|<\infty$, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, the...

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the...

For a function $f\in L_{loc}^p\left(ℝⁿ\right)$ the notion of p-mean variation of order 1, ${₁}^p(f,ℝⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1,p}\left(ℝⁿ\right)$ in terms of ${₁}^p(f,ℝⁿ)$ is directly related to the characterisation of $W^{1,p}\left(ℝⁿ\right)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

To each set of knots $t_i=i/2n$ for i = 0,...,2ν and $t_i=(i-\nu )/n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space ${}_{\nu ,n}$ of all piecewise linear and continuous functions on I = [0,1] with knots $t_i$ and the orthogonal projection $P_{\nu ,n}$ of L²(I) onto ${}_{\nu ,n}$. The main result is $li{m}_{(n-\nu )\wedge \nu \to \infty }\left|\right|P_{\nu ,n}\left|\right|₁=su{p}_{\nu ,n:1\le \nu \le n}\left|\right|P_{\nu ,n}\left|\right|₁=2+(2-\surd 3)²$. This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1<p<\infty$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $1\le p<\infty$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a “common roof”...

Let ${\left(F_n\right)}_{n\in \mathbb{N}}$ be a sequence of non-decreasing functions from $[0,+\infty )$ into $[0,+\infty )$. Under some suitable hypotheses of ${\left(F_n\right)}_{n\in \mathbb{N}}$, we will prove that if $g\in L^p\left({ℝ}^N\right)$, $1<p<+\infty$, satisfies ${lim\; inf}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy<+\infty$, then $g\in W^{1,p}\left({ℝ}^N\right)$ and moreover ${lim}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy=K_{N,p}{\int }_{{ℝ}^N}{\left|\nabla g\left(x\right)\right|}^{p}dx$, where $K_{N,p}$ is a positive constant depending only on $N$ and $p$. This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some...

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections $P_a$ determined by the different involutions ${}_a$ induced by positive invertible elements a ∈ A. The maps $\phi :P\to P_a$ sending p to the unique $q\in P_a$ with the same range as p and ${\Omega }_a:P_a\to P_a$ sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| <...

Let $\Omega \subset {ℝ}^n,n\ge 3$, and let $p,1<p<\infty ,p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega$, in the setting of $\delta$-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta }=\tilde{\delta }(p,n)>0$ small such that if $\Omega$ is a $\delta$-Reifenberg flat domain with $\delta <\tilde{\delta }$, then $p$-harmonic measure is concentrated on a set of $\sigma$-finite $H^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic...

Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^1$-base and data in $h_c^1$, a subspace of $L$1. We derive our results, considering the action of an adjoint operator on $B_{T}MOC$, a predual of $h_c^1$, and using known properties of this last space.

In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $𝔻$, if $F\left(𝔻\right)$ is a convex domain, then the inequality $|G\left(z_2\right)-G\left(z_1\right)|<|H\left(z_2\right)-H\left(z_1\right)|$ holds for all distinct points $z_1,z_2\in 𝔻$. Here $H$ and $G$ are holomorphic mappings in $𝔻$ determined by $F=H+\overline{G}$, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $\Omega$ in $ℂ$ and improve it provided $F$ is additionally a quasiconformal mapping...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

It is shown that there is a subspace $Z_q$ of ${\ell }_q$ for $1<q<2$ which is isomorphic to ${\ell }_q$ such that ${\ell }_q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty$ there is a subspace $Y_p$ of ${\ell }_p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space ${\ell }_p/Y_p$ is isomorphic to ${\ell }_p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty$ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to ${\ell }_p$.

CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................

Let $G$ be any group containing an infinite elementary amenable subgroup and let $2\&lt;p\&lt;\infty$. We construct an exhaustion of ${\ell }^{p}G$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to ${\ell }^p$-dimension and gives an answer to a question of Gaboriau.

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺. ...

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...