Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \frac1j(N) ? _{0 ￡ m < Ngcd(m,N)=1} |S(m,N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form A_h(Q)=\frac1?_{\fracaq ? FQ}h(\fracaq) ×?_{\fracaq ? FQ}h(\fracaq) |s(a^￠,q^￠)-s(a,q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert , where h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C} is a continuous function with ò₀¹ h(t)  d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0 , \fracaq{\frac{a}{q}} runs over F_Q{\cal F}_{\!Q} , the set of Farey fractions of order Q in the unit interval [0,1] and \fracaq < \fraca^￠q^￠{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}} are consecutive elements of F_Q{\cal F}_{\!Q} . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

On Approximate Spectral Factorization of Matrix Functions

It is proved that if positive definite matrix functions (i.e. matrix spectral densities) S _n , n=1,2,… , are convergent in the L ₁-norm, ||S_n-S||_L1? 0\|S_{n}-S\|_{L_{1}}\to 0, and ò₀^2plogdetS_n(e^iq) dq?ò₀^2plogdetS(e^iq) dq\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S_{n}(e^{i\theta})\,d\theta\to\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S(e^{i\theta})\,d\theta, then the corresponding (canonical) spectral factors are convergent in L ₂, ||S⁺_n-S⁺||_L2? 0\|S^{+}_{n}-S^{+}\|_{L_{2}}\to 0. The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place.     

A separation theorem and Serre duality for the Dolbeault cohomology

LetX be a complex manifold with finitely many ends such that each end is eitherq-concave or (n−q)-convex. If , then we prove thatH ^pn−q (X) is Hausdorff for allp. This is not true in general if (Rossi’s example withn=2 andq=1). If all ends areq-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for ). Moreover the result was already known in the case when theq-concave ends can be ‘filled in’ (again also for ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries, . This research was partially supported by TMR Research Network ERBFMRXCT 98063.     

Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s]⁽ⁿ⁺¹⁾E[s]     \text as   n?￥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

We study the well-posedness of the fractional differential equations with infinite delay (P ₂): D^a u(t)=Au(t)+ò^t_-￥a(t-s)Au(s)ds + f(t), (0 ￡ t ￡ 2p){D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}, where A is a closed operator in a Banach space ${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)} and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P ₂) on Lebesgue-Bochner spaces L^p(\mathbbT, X){L^p(\mathbb{T}, X)} and periodic Besov spaces B _p,q^s(\mathbbT, X){{B} _{p,q}^s(\mathbb{T}, X)} .     

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L^\fracn2(\mathbbRⁿ){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L ^q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K₁(K₂|b|)^|b|t^{-\frac|b|2-1+\fracn2q}{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K ₁ and K ₂. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]^-2+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

On certain functional equations related to Jordan triple (q, f){(\theta, \phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Q_s{q\in Q_{s}}.     

Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces

The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space ${\Upsilon\colon\widehat X\to X}The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space U\colon[^(X)]? X{\Upsilon\colon\widehat X\to X} in which [^(Y)] o U^-1(Y){\widehat Y\equiv\Upsilon^{-1}(Y)} has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C ^∞ exhaustion function j{\varphi} on [^(X)]{\widehat X} which is strongly q-convex on [^(W)]=U^-1(W){\widehat\Omega=\Upsilon^{-1}(\Omega)} outside a uniform neighborhood of the q-dimensional compact irreducible components of [^(Y)]{\widehat Y}.     

Localization and Locality for Resistance Forms

For a resistance form ${(X, \mathcal{D}(\varepsilon),\varepsilon)}For a resistance form (X, D(e),e){(X, \mathcal{D}(\varepsilon),\varepsilon)} and a point x₀ ? X{x_0 \in X} as boundary, on the space X₀:=X \{x₀}{X_0:=X {\setminus}\{x_0\}} we consider the Dirichlet space D_x0:={f ? D(e) | f(x₀)=0}{\mathcal{D}_{x_0}:=\{f\in\mathcal{D}(\varepsilon)\, |\, f(x_0)=0\}} and we develop a good potential theory. For any finely open subset D of X ₀ we consider a localized resistance form (D_{X0 \ D},e_D{\mathcal{D}_{X_0 {\setminus} D},\varepsilon_{D}}) where D_{X0 \ D}:={f ? D_x0 | f=0{\mathcal{D}_{X_0 {\setminus} D}:=\{f\in\mathcal{D}_{x_0}\, |\, f=0} on X₀ \ D}, e_D(f,g):=e(f,g){X_0 {\setminus} D\},\, \varepsilon_D(f,g):=\varepsilon(f,g)} for all f,g ? D_{X0 \ D}{f,g\in\mathcal{D}_{X_0 {\setminus} D}}. The main result is the equivalence between the local property of the resistance form and the sheaf property for the excessive elements on finely open sets.     

A characterisation result on a particular class of non-weighted minihypers

We present a characterisation of {e₁ (q+1)+e₀,e₁ ;n,q}{\{\epsilon_1 (q+1)+\epsilon_0,\epsilon_1 ;n,q\}} -minihypers, q square, q = p ^h , p > 3 prime, h ≥ 2, q ≥ 1217, for e₀ + e₁ < \fracq^7/122-\fracq^1/42{\epsilon_0 + \epsilon_1 < \frac{q^{7/12}}{2}-\frac{q^{1/4}}{2}}. This improves a characterisation result of Ferret and Storme (Des Codes Cryptogr 25(2): 143–162, 2002), involving more Baer subgeometries contained in the minihyper.     

On the sum of a prime and a k-th power of prime in short intervals

Let H_k\mathcal{H}_{k} denote the set {n∣2|n, n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when X^{\frac1120(1-\frac12k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n ? \allowbreak H_k ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X^{\frac1120(1-\frac1k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.     

Some estimates of Schrödinger type operators on the Heisenberg group

In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D_{\mathbb H}ⁿ)² +V ²{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class B_q1 for q₁ 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D_{\mathbb Hⁿ}{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hⁿ{\mathbb {H}^n}. An L ^p estimate and a weak type L ¹ estimate for the operator ?⁴_{\mathbb Hⁿ} H^-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? B_q1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq₁2{1 < p \leq \frac{q_{1}}{2}} are obtained.     

Markoff–Lagrange spectrum and extremal numbers

Let \( \gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right) \) denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant c > 0 with the property that, for arbitrarily large values of X, the inequalities\( \left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}} \)admit no non-zero solution \( \left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3} \). Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X ≥ 1. Such extremal numbers are transcendental and their set is stable under the action of \( {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) \) on \( \mathbb{R}\backslash \mathbb{Q} \) by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf _q→∞ q||qξ|| is \( \frac{1}{3} \), the largest possible value for a non-quadratic number, and that there is a natural bijection between the \( {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) \)-equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Complex Interpolation of Operators and Optimal Domains

Let X ₀ and X ₁ be two order continuous Banach function spaces on a finite measure space, (E ₀, E ₁) a Banach space interpolation pair, and \({T: X_0 + X_1 \to E_0 + E_1}\) an admissible operator between the pairs (X ₀,X ₁) and (E ₀,E ₁). If \({T_{\theta} : [X_0, X_1]_{[\theta ]} \to [E_0, E_1]_{[\theta]}}\) is the interpolated operator by the first complex method of Calderón and m ₀, m ₁ and m _θ are the vector measures coming from \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) and T _θ, respectively, then we study the relationship between the optimal domain \({L^1(m_{\theta})}\) of T _θ and the complex interpolation space \({[L^1(m_0),L^1(m_1)]_{[\theta]}}\) of the optimal domains of \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) . Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p,q)-power-concave operators.     

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

A class U_k1 (J){\mathcal{U}}_{\kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q  mvf￠s S_k^{p ×q}p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation T_W based on W and applied to S_k2^{p ×q}{\mathcal{S}}_{\kappa 2}^{p \times q}. Factorization formulas for mvf’s W in a subclass U^°_k1 (J) of U_k1(J){\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J) found and then used to parametrize the set S_k1+k2^{p ×q} ?T_W [ S_k2^{p ×q} ]{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]. Applications to bitangential interpolation problems in the class S_k1+k2^{p ×q}{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} will be presented elsewhere.     

A regularity criterion for axially symmetric solutions to the Navier–Stokes equations

The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v _r ) of the velocity belongs to L _∞(0, T;L ₃(Ω₀)) or v _r /r belongs to L _∞(0, T;L _3/2(Ω₀)), where Ω₀ is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω _k , k = 1, . . . , N, such that W₀ ì è_{k = 1}^N W_k {\Omega_0} \subset \bigcup\limits_{k = 1}^N {{\Omega_k}} , and assume that there exist constants α ₁, α ₂ such that either || v_r ||_{L￥ ( 0,T;L3( Wk ) )} ￡ a₁ or  || \fracv_rr ||_{L￥ ( 0,T;L3/2( Wk ) )} ￡ a₂ {\left\| {{v_r}} \right\|_{{L_\infty }\left( {0,T;{L_3}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_1}\,or\;{\left\| {\frac{{{v_r}}}{r}} \right\|_{{L_\infty }\left( {0,T;{L_{3/2}}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W₂^2,1( W×( 0,T ) ),?p ? L₂( W×( 0,T ) ) ) \left( {v \in W_2^{2,1}\left( {\Omega \times \left( {0,T} \right)} \right),\nabla p \in {L_2}\left( {\Omega \times \left( {0,T} \right)} \right)} \right) . Bibliography: 28 titles.