Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

Some classes of pre-Hilbert algebras with norm-one central idempotent

In this paper we first characterize the pre-Hilbert algebras with a norm-one central idempotent e such that ‖ex‖ = ‖x‖ for any x ∈ A. This generalizes a well-known theorem by Ingelstam asserting that every alternative pre-Hilbert algebra with a unit 1 such that ‖1‖ = 1 is isomorphic to ?, ?, ? or $\mathbb{O}$ . We also show that every power-associative pre-Hilbert algebra satisfying ‖x ²‖ = ‖x‖² for every element has a unique nonzero idempotent, which is a unit element. In fact, the same conclusion will be proved in a more general setting. As application we give some conditions characterizing when a real algebra A, which is a prehilbert space, is isomorphic to one of the Hilbert algebras ?, ?, ? or $\mathbb{O}$ .     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

On the similarity problem for polynomially bounded operators on hilbert space

Partial solutions are obtained to Halmos’ problem, whether or not any polynomially bounded operator on a Hilbert spaceH is similar to a contraction. Central use is made of Paulsen’s necessary and sufficient condition, which permits one to obtain bounds on ‖S‖ ‖S ^?1‖, whereS is the similarity. A natural example of a polynomially bounded operator appears in the theory of Hankel matrices, defining $$R_f = \left( {\begin{array}{*{20}c} {S*} \\ 0 \\ \end{array} \begin{array}{*{20}c} {\Gamma _f } \\ S \\ \end{array} } \right)$$ onl ² ⊕l ², whereS is the shift and Γ _f the Hankel operator determined byf withf′ ∈ BMOA. Using Paulsen’s condition, we prove thatR _f is similar to a contraction. In the general case, combining Grothendieck’s theorem and techniques from complex function theory, we are able to get in the finite dimensional case the estimate $$\left\| S \right\|\left\| {S^{ - 1} } \right\| \leqq M^4 log(dim H)$$ whereSTS ^?1 is a contraction and assuming \(\left\| {p\left( T \right)} \right\| \leqq M\left\| p \right\|_\infty \) wheneverp is an analytic polynomial on the disc.     

Convergence of the Vallée-Poussin means for Fourier-Jacobi sums

Letf εC[?1, 1], ?1<α,β≤0, let $f \in C[ - 1, 1], - 1< \alpha , \beta \leqslant 0$ , letS _n ^{α, β} (f, x) be a partial Fourier-Jacobi sum of ordern, and let $$\nu _{m, n}^{\alpha , \beta } = \nu _{m, n}^{\alpha , \beta } (f) = \nu _{m, n}^{\alpha , \beta } (f,x) = \frac{1}{{n + 1}}[S_m^{\alpha ,\beta } (f,x) + ... + S_{m + n}^{\alpha ,\beta } (f,x)]$$ be the Vallée-Poussin means for Fourier-Jacobi sums. It was proved that if 0<a≤m/n≤b, then there exists a constantc=c(α, β, a, b) such that ‖ν _{m, n} ^{α, β} ‖ ≤c, where ‖ν _{m, n} ^{α, β} ‖ is the norm of the operator ν _{m, n} ^{α, β} inC[?1,1].     

Small into-isomorphism from L∞ (A, μ) into L∞(B, υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ^?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.     

Exact constants in generalized inequalities for intermediate derivatives

Consider the Sobolev space W ₂ ⁿ (?₊) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants A _n,k in inequalities of Kolmogorov type for the values of intermediate derivatives |f ^(k)(0)| ≤ A _n,k ‖f‖. In the general case, the expression for the constants A _n,k is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants A _n,k in the case of the following norms: $$ \left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } . $$ In the case of the norm ‖ · ‖₁, formulas for the constants A _n,k were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants A _n,k is also studied in the case of the norm ‖ · ‖₂. In addition, we prove a symmetry property of the constants A _n,k in the general case.     

Equiconvergence of spectral decompositions of Hill operators

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = ?d ²/dx ² + v(x), x ∈ L ¹([0, π], with H _per ^?1 -potential and the free operator L ⁰ = ?d ²/dx ², subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $\left\| {S_N - S_N^0 :L^a \to L^b } \right\| \to 0if1 < a \leqslant b < \infty ,1/a - 1/b < 1/2,$ , where S _N and S _N ⁰ are the N-th partial sums of the spectral decompositions of L and L ⁰. Moreover, if v ∈ H ^?α with 1/2 < α < 1 and $\frac{1} {a} = \frac{3} {2} - \alpha $ , then we obtain the uniform equiconvergence ‖S _N ?S _N ⁰ : L ^a → L ^∞‖ → 0 as N → ∞.     

On the class of limits of lacunary trigonometric series

Let (n _k ) _k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n _k+1/n _k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝ ₀ ¹ f(x) dx = 0. Then the probabilistic behavior of the system (f(n _k x)) _k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erd?s and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary $ (n_k )_{k \geqq 1} $ : (1) $$ \mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2 $$ for almost all x ∈ (0, 1), where ‖f‖₂ = (∝ ₀ ¹ f(x)² dx)^1/2 is the standard deviation of the random variables f(n _k x). If (n _k ) _k≧1 has certain number-theoretic properties (e.g. n _k+1/n _k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f‖₂. For general lacunary (n _k ) _k≧1 this is not necessarily true: Erd?s and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n _k ) _k≧1, such that the lim sup in the LIL (1) is not equal to ‖f‖₂ and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (n _k ) _k≧1 such that (1) holds with √‖f‖ ₂ ² + g(x) instead of ‖f‖₂ on the right-hand side.     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

A unifying approach to the regularization of Fourier polynomials

In a previous paper [4] the following problem was considered:find, in the class of Fourier polynomials of degree n, the one which minimizes the functional: (0.1) $$J^* [F_n ,\sigma ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\frac{{\sigma ^r }}{{r!}}} \left\| {F_n^{(r)} } \right\|^2$$ , where ∥·∥ is theL ² norm,F _n ^(r) is therth derivative of the Fourier polynomialF _n (x), andf(x) is a given function with Fourier coefficientsc _k . It was proved that the optimal polynomial has coefficientsc _k ^* given by (0.2) $$c_k^* = c_k e^{ - \sigma k^2 } ; k = 0, \pm ,..., \pm n$$ . In this paper we consider the more general functional (0.3) $$\hat J[F_n ,\sigma _r ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\sigma _r \left\| {F_n^{(r)} } \right\|^2 }$$ , which reduces to (0.1) forσ _r =σ ^r /r!. We will prove that the classical sigma-factor method for the regularization of Fourier polynomials may be obtained by minimizing the functional (0.3) for a particular choice of the weightsσ _r . This result will be used to propose a motivated numerical choice of the parameterσ in (0.1).     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation

It is shown that the uniform distance between the distribution function $F_n^K(h)$ of the usual kernel density estimator (based on an i.i.d. sample from an absolutely continuous law on ${\mathbb{R}}$ ) with bandwidth h and the empirical distribution function F _n satisfies an exponential inequality. This inequality is used to obtain sharp almost sure rates of convergence of $\|F_n^K(h_n)-F_n\|_\infty$ under mild conditions on the range of bandwidths h _n , including the usual MISE-optimal choices. Another application is a Dvoretzky–Kiefer–Wolfowitz-type inequality for $\|F_n^{K}(h)-F\|_\infty$ , where F is the true distribution function. The exponential bound is also applied to show that an adaptive estimator can be constructed that efficiently estimates the true distribution function F in sup-norm loss, and, at the same time, estimates the density of F—if it exists (but without assuming it does)—at the best possible rate of convergence over Hölder-balls, again in sup-norm loss.     

Roots of unity and the polynomials with coefficients in{\mathbb{T}}

The maximal order of the coefficients of an arbitrary polynomialP having coefficients in ${\mathbb{T}}$ and the quotient of ‖P‖ _L ⁴ divided by ‖P‖ _L ² depends on the behavior of them in ${\mathbb{T}}$ . In this paper we show thatP can be approximated with another such polynomials which has coefficients as roots of unity.     

Wavelet characterization of growth spaces of harmonic functions

We consider the space h _ν ^∞ of harmonic functions in R ₊ ⁿ⁺¹ with finite norm ‖u‖ _ν = sup |u(x, t)|/v(t), where the weight ν satisfies the doubling condition. Boundary values of functions in h _ν ^∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces h _ν ^∞ ～ l ^∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in h _ν ^∞ along vertical lines.     

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations

The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds.     

w* sequential convergence

LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ _n ^* } _n=1 ^∞ ?X* which arew* converging to 0 while inf _n ‖x* _n ‖>0, are constructed.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.