A generalized Khintchine inequality in rearrangement invariant spaces

Let X be a separable or maximal rearrangement invariant space on [0, 1]. Necessary and sufficient conditions are found under which the generalized Khintchine inequality

$\left\| {\sum\limits_{k = 1}^\infty {f_k } } \right\|_X \leqslant C\left\| {\left( {\sum\limits_{k = 1}^\infty {f_k^2 } } \right)^{1/2} } \right\|_X $

holds for an arbitrary sequence {?_k} _k=1 ^∞ ? X of mean zero independent variables. Moreover, the subspace spanned in a rearrangement invariant space by the Rademacher system with independent vector coefficients is studied.

     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

A note on skew characters of symmetric groups

In this paper we establish the following estimate:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$

where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log⁺(t)). This inequality relies upon the following sharp L ^p estimate:

$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$

where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$

We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.

     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.     

Some Berezin Number Inequalities for Operator Matrices

The Berezin symbol à of an operator A acting on the reproducing kernel Hilbert space H = H(Ω) over some (nonempty) set is defined by \(\tilde A(\lambda ) = \left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle \), λ ∈ Ω, where \(\hat k_\lambda = k_\lambda /\left\| {k_\lambda } \right\|\) is the normalized reproducing kernel of H. The Berezin number of the operator A is defined by \(ber(A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle } \right|\). Moreover, ber(A) ? w(A) (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if \(T = \left[ {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right] \in \mathbb{B}(\mathcal{H}(\Omega _1 ) \oplus \mathcal{H}(\Omega _2 ))\), then

$$ber(T) \leqslant \frac{1}{2}(ber(A) + ber(D)) + \frac{1}{2}\sqrt {(ber(A) - ber(D))^2 + \left( {\left\| B \right\| + \left\| C \right\|} \right)^2 } .$$

     

Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

We study the operator-valued positive dyadic operator

$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$

where the coefficients {λ _Q : C → D} _Q∈D are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.

In the two-weight case, we prove that the L _C ^p (σ) → L _D ^q (ω) boundedness of the operator T _λ( · σ) is characterized by the direct and the dual L ^∞ testing conditions:

$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$

,

$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$

.

Here L _C ^p (σ) and L _D ^q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.

In the unweighted case, we show that the L _C ^p (μ) → L _D ^p (μ) boundedness of the operator T _λ( · μ) is equivalent to the end-point direct L ^∞ testing condition:

$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$

.

This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

Weak Uniform Normal Structure and Fixed Points of Asymptotically Regular Semigroups

Let X be a Banach space with a weak uniform normal structure and C a non–empty convexweakly compact subset of X. Under some suitable restriction, we prove that every asymptoticallyregular semigroup T = {T(t) : t ∈¸ S} of selfmappings on C satisfying

${\mathop {\lim \inf }\limits_{S \mathrel\backepsilon t \to \infty } }{\left| {{\left\| {T(t)} \right\|}} \right|} < {\text{WCS}}(X)$

has a common fixed point, where WCS(X) is the weakly convergent sequence coefficient of X, and\({\left| {{\left\| {T(t)} \right\|}} \right|}\) is the exact Lipschitz constant of T(t).     

On a property of the reducibility exponent of linear differential systems

We establish that the reducibility exponent (Differentsial’nye Uravneniya, 2007, vol. 43, no. 2, pp. 191–202) of each linear system

$$\dot x = A(t)x, x \in \mathbb{R}^n , t \geqslant 0$$

, with piecewise continuous bounded coefficient matrix A does not belong to the set of values of σ for which the perturbed system (1_A+Q) with an arbitrary piecewise continuous perturbation Q satisfying the condition \(\overline {\lim } _{t \to + \infty } t^{ - 1} \ln \left\| {Q(t)} \right\| \leqslant - \sigma \) is reducible to the original system (1 _A ) by some Lyapunov transformation.

     

On spectral estimates for Schrödinger-type operators: The case of small local dimension

The behavior of the discrete spectrum of the Schrödinger operator - Δ -V is determined to a large extent by the behavior of the corresponding heat kernel P(t; x,y) as t → 0 and t→ ∞. If this behavior is power-like, i.e.,

$\left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - \delta /2} ),t \to 0,\left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - D/2} ),t \to \infty ,$

then it is natural to call the exponents δ and D the local dimension and the dimension at infinity, respectively. The character of spectral estimates depends on a relation between these dimensions. The case where δ < D, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.

     

Some remarks on Wente’s inequality and the Lorentz-Sobolev space

In this note we consider Wente's type inequality on the Lorentz-Sobolev space.If▽f∈L~p1,q1(R~n),G ∈ L~(p2,q2)(R~n) and div G≡0 in the sense of distribution where(1/p1)+(1/P2)=(1/q1)+(1/q2)=1,1P1,P2∞,it is known that G·▽f belongs to the Hardy space H~1 and furthermore‖G·▽f‖H~1≤C‖▽f‖L~(p1,q1)(R~2)‖G‖L~(p2,q2)(R~2).Reader can see[9]Section 4.Here we give a new proof of this result.Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest.Finally,we use this inequality to get a generalisation of Bethuel's inequality.     

Complementary inequalities to improved AM-GM inequality

Following an idea of Lin, we prove that if A and B are two positive operators such that 0 mI ≤ A ≤m'I≤ M'I ≤ B ≤ MI, then Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2Φ~2(A≠B) and Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2(Φ(A)≠Φ(B))~2 where K(h)=(h+1)~2/4 and h = M/m and Φ is a positive unital linear map.     

On the numerical index of real <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(µ)-spaces

We give a lower bound for the numerical index of the real space L _p (µ) showing, in particular, that it is non-zero for p ≠ 2. In other words, it is shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {|\int {|x{|^{p - 1}}{\rm{sign}}(x)Tx d\mu |:x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {{{M_p}} \over {12{\rm{e}}}}||T||$

where \({M_p} = {\max _{t \in \left[ {0,1} \right]}}{{|{t^{p - 1}} - t|} \over {1 + {t^p}}} > 0\) for every p ≠ 2. It is also shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {\int {|x{|^{p - 1}}|Tx| d\mu :x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {1 \over {2{\rm{e}}}}||T||$

.

     

Precise asymptotics for complete moment convergence in Hilbert spaces

Let {X, X _n ; n?≥?1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space \((\textbf{H},\|\cdot\|)\) with covariance operator Σ. Set \(S_n=\sum_{i=1}^nX_i,\) n?≥?1. We prove that for 1?p?r?>?1?+?p/2,

$\begin{array}{lll} &;\lim\limits_{\varepsilon\searrow0}\varepsilon^{(2r-p-2)/(2-p)}\sum\limits_{n=1}^\infty n^{r/p-2-1/p}{\mbox{\rm{\textsf{E}}}}\{\|S_n\|-\sigma\varepsilon n^{1/p}\}_+\\&;\quad\qquad\qquad\qquad=\sigma^{-(2r-2-p)/(2-p)}\frac{p(2-p)}{(r-p)(2r-p-2)}{\mbox{\rm{\textsf{E}}}}\|Y\|^{2(r-p)/(2-p)}, \end{array}$

where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, and σ ² is the largest eigenvalue of Σ.

     

Conditional exponential stability of a differential system with linear dichotomous Coppel-Conti approximation

We prove the conditional exponential stability of the zero solution of the nonlinear differential system

$$\dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0,$$

with L ^p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X _A (t), X _A (0) = E, satisfies the condition

$$\mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0,$$

where P ₁ and P ₂ are complementary projections of rank k ∈ {1, …, n ? 1} and rank n ? k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R ⁿ that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin.

     

Coverings of random ellipsoids,and invertibility of matrices with i.i.d. heavy-tailed entries

Let A = (a_ij) be an n × n random matrix with i.i.d. entries such that Ea₁₁ = 0 and Ea ₁₁ ² = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B ₂ ⁿ of cardinality at most exp(δn) such that with probability very close to one we have

$$A\left( {B_2^n} \right)\subset\mathop \cup \limits_{y \in A\left( \mathcal{N} \right)} \left( {y + L\sqrt n B_2^n} \right)$$

. In fact, a stronger statement holds true. As an application, we show that for some L' > 0 and u ∈ [0, 1) depending only on the distribution law of a11, the smallest singular value sn of the matrix A satisfies

$$\mathbb{P}\left\{ {{s_n}\left( A \right) \leq \varepsilon {n^{ - 1/2}}} \right\} \leq L'\varepsilon + {u^n}$$

for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

     

Relations between the norms of a function and its gradient in classes of surface and solid spherical harmonics on a finite-dimensional space

Relations between the Euclidean (mean-square) and Chebyshev (uniform) norms of a surface spherical harmonic of order n and its gradient in the tangent bundle of the sphere \(\mathbb{S}^{k - 1} \), as well as relations between the same norms of a solid spherical harmonic (internal and external) of order n and its spatial gradient, are given. In all cases, the norm of the gradient differs from that of the harmonic by a factor of order n. As examples, relations between the norms of the internal and external solid spherical harmonics

$r\left\langle {gradU_n } \right\rangle = n\left\langle {U_n } \right\rangle , r\left\| {gradU_n } \right\| = \sqrt {n(2n + k - 2)} \left\| {U_n } \right\|$

and

$r\left\langle {gradV_n } \right\rangle = (n + k - 2)\left\langle {V_n } \right\rangle , r\left\| {gradV_n } \right\| = \sqrt {(n + k - 2)(2n + k - 2)} \left\| {V_n } \right\|,$

respectively, are considered; here, <·> and ∥·∥ denote the Chebyshev and Euclidean norms, respectively.

     

Isometries with dense windings of the torus in <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">M</Emphasis>)

Let C(M) be the space of all continuous functions on M? ?. We consider the multiplication operator T: C(M) → C(M) defined by Tf(z) = zf(z) and the torus

$$O(M) = \left\{ {f:M \to \mathbb{C} \ntrianglelefteq \left\| f \right\| = \left\| {\frac{1}{f}} \right\| = 1} \right\}$$

. If M is a Kronecker set, then the T-orbits of the points of the torus ½O(M) are dense in ½O(M) and are ½-dense in the unit ball of C(M).

     

Two wide neighborhood interior-point methods for symmetric cone optimization

In this paper, we present two primal–dual interior-point algorithms for symmetric cone optimization problems. The algorithms produce a sequence of iterates in the wide neighborhood \(\mathcal {N}(\tau ,\,\beta )\) of the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We derive that these two path-following algorithms have

$$\begin{aligned} \text{ O }\left( \sqrt{r\text{ cond }(G)}\log \varepsilon ^{-1}\right) , \text{ O }\left( \sqrt{r}\left( \text{ cond }(G)\right) ^{1/4}\log \varepsilon ^{-1}\right) \end{aligned}$$

iteration complexity bounds, respectively. The obtained complexity bounds are the best result in regard to the iteration complexity bound in the context of the path-following methods for symmetric cone optimization. Numerical results show that the algorithms are efficient for this kind of problems.

     

How f ar is $${\left( 1 + \fr ac{ a}{n}\right) ^n} $$1+ ann from $$ e^ a $$e a when a is an element of a B an ach algebr a?

We present effective upper and lower bounds for the distance from \(\displaystyle \left( 1 + \frac{a}{n}\right) ^n \) to \(\displaystyle e^a \) for an element a of a complex unital Banach algebra and positive integer n. Specifically

$$\begin{aligned} \frac{1}{2n} \sup \left\{ \left| \mathfrak {R}(z^2) \right| e^{\mathfrak {R}(z)} : z \in \sigma (a) \right\} \lesssim _{(2)} \left\| e^a - \left( 1 + \frac{a}{n}\right) ^n \right\| \le \frac{ \left\| a \right\| ^2}{2n} \ e^{ \left\| a \right\| }, \end{aligned}$$

where \(\sigma (a)\) is the spectrum of a. The symbol \(\lesssim _{(p)}\) means “less than or equal to, up to a term of order \(n^{-p}\)”as discussed below. Following some technical preliminaries (Sect. 1) we treat the real case (Sect.  2), extend to the complex case (Sect.  3), and then generalise to the case of a norm-unital Banach algebra (Sect.  4).