Sharp bounds for integral means

We introduce a bound M of f, ‖f‖_∞?M?2‖f‖_∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|^p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule.     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

Ill-posed problems with unbounded operators

Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au=f is solvable, and ‖fδ−f‖?δ, then the following results are provided: Problem Fδ(u):=‖Au−fδ^‖2+α‖u^‖2 has a unique global minimizer uα,δ for any fδ, uα,δ=A^*−1(AA^*+αI)fδ. There is a function α=α(δ), limδ→0α(δ)=0 such that limδ→0‖uα(δ),δ−y‖=0, where y is the unique minimal-norm solution to (1). A priori and a posteriori choices of α(δ) are given. Dynamical Systems Method (DSM) is justified for Eq. (1).     

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.     

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.     

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.     

On a new notion of the solution to an ill-posed problem

A new understanding of the notion of the stable solution to ill-posed problems is proposed. The new notion is more realistic than the old one and better fits the practical computational needs. A method for constructing stable solutions in the new sense is proposed and justified. The basic point is: in the traditional definition of the stable solution to an ill-posed problem Au=f, where A is a linear or nonlinear operator in a Hilbert space H, it is assumed that the noisy data {f_δ,δ} are given, ‖f−f_δ‖≤δ, and a stable solution u_δ:=R_δf_δ is defined by the relation lim_δ→0‖R_δf_δ−y‖=0, where y solves the equation Au=f, i.e., Ay=f. In this definition y and f are unknown. Any f∈B(f_δ,δ) can be the exact data, where B(f_δ,δ):={f:‖f−f_δ‖≤δ}.The new notion of the stable solution excludes the unknown y and f from the definition of the solution. The solution is defined only in terms of the noisy data, noise level, and an a priori information about a compactum to which the solution belongs.     

On the effective determination of Maass cusp forms by L-values

Let f be a Hecke–Maass cusp form of Laplace eigenvalue 1/4+μ ² with |μ|≤Λ for \(\mathit{SL}_{2}(\mathbb{Z})\) . We show that f is uniquely determined by the central values of Rankin–Selberg L-functions L(s,f?g), where g runs over the set of holomorphic cusp forms of weight k? _? Λ ^1+3θ+? for any ?>0 for \(\mathit{SL}_{2}(\mathbb{Z})\) .     

The mean square of Dirichlet series associated with automorphic forms

Letf be a non-holomorphic automorphic form of real weight and eigenvalue λ=1/4?ρ ², ?ρ≥0, which is defined with respect to a Fuchsian group of the first kind. Assume that ∞ is a cusp of this group and denote bya _∞,n,a _∞,n ,n ∈ ?, the Fourier coefficients off at ∞. Following Hecke and Maas we prove that under suitable assumptions the associated Dirichlet seriesL ⁺ (f, s) = ∑ _{n > 0} a _∞,n (n + μ_221E;)^?s andL ^? (f, s) = ∑ _{n < 0} a _∞,n |n + μ_221E;|^?s have meromorphic continuation in the entire complex plane and statisfy a certain functional equation (μ_∞ denotes the cusp parameter of the cusp ∞). We are interested in mean square estimates of these functions. Iff is not a cusp form we prove $$\int_0^T {|L^ \pm (f,\Re _\rho + it)|^2 dt = T(\log T)^a (B^ \pm + o(1)),}$$ wherea is either 1, 2 or 4, andB ^± is a constant. A similar result is true iff is a cusp form. In case of a congruence group the termo(1) can be replaced byO ((logT)^?1).     

Weighted Weierstrass' theorem with first derivatives

We characterize the set of functions which can be approximated by continuous functions with the norm ‖f_‖L^∞(w) for every weight w. This fact allows to determine the closure of the space of polynomials in L^∞(w) for every weight w with compact support. We characterize as well the set of functions which can be approximated by smooth functions with the norm

‖f_‖W1,∞(w₀,w₁):=‖f_‖L^∞(w₀)+‖f^′_‖L^∞(w₁),     

Isoperimetric type inequalities for harmonic functions

For 0<p<+∞ let hp be the harmonic Hardy space and let bp be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1?p<+∞, denote by ‖⋅bp_‖ and ‖⋅hp_‖ the norms in the spaces bp and hp, respectively. In this paper, we establish the harmonic hp-analogue of the known isoperimetric type inequality ‖fb2p_‖?‖fhp_‖, where f is an arbitrary holomorphic function in the classical Hardy space Hp. We prove that for arbitrary p>1, every function f∈hp satisfies the inequality

‖fb2p_‖?ap‖fhp_‖,     

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <p < ∞ and μ > 0. ThenL has a bounded H_∞ functional calculus in L^p(Ω), in the sense that ¦¦f (L +cI)u¦¦_p ≤C sup_¦arλ¦<μ ^¦f¦ ¦‖u¦‖_p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderón-Zygmund singular integral operators.     

Kolmogorov-type inequalities for norms of Riesz derivatives of functions of several variables with Laplacian bounded in L ∞ and related problems

Let L _∞,∞ ^Δ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that Δf ∈ L _∞(? ^m ). We obtain new sharp Kolmogorov-type inequalities for the L _∞-norms of the Riesz derivatives D ^α f of the functions f ∈ L _∞,∞ ^Δ (? ^m ) and solve the Stechkin problem of approximating an unbounded operator D ^α by bounded operators on the class f ∈ L _∞(? ^m ) such that ‖Δf‖_∞ ≤ 1, and also the problem of the best recovery of the operator D ^α from elements of this class given with error δ.     

Bounds for norms of the matrix inverse and the smallest singular value

In the first part, we obtain two easily calculable lower bounds for ‖A^-1‖, where ‖·‖ is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for ‖A^-1‖_∞ and ‖A^-1‖₁ in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for ‖A^-1‖₁ in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.     

Uniform separation of points and measures and representation by sums of algebras

LetX andY _i, 1 ≦i ≦k, be compact metric spaces, and letρ _i:X →Y _i be continuous functions. The familyF={ρ _i} _i ^1/k is said to be ameasure separating family if there exists someλ > 0 such that for every measureμ inC(X)*, ‖μ ^o ρ _i ^?1 ‖ ≧λ ‖μ ‖ holds for some 1 ≦i ≦k.F is auniformly (point) separating family if the above holds for the purely atomic measures inC(X)*. It is known that fork ≦ 2 the two concepts are equivalent. In this note we present examples which show that fork ≧ 3 measure separation is a stronger property than uniform separation of points, and characterize those uniformly separating families which separate measures. These properties and problems are closely related to the following ones: letA ₁,A ₂, ...,A _k be closed subalgebras ofC(X); when isA ₁ +A ₂ + ... +A _k equal to or dense inC(X)?     

Resonance between automorphic forms and exponential functions

Let f be a holomorphic cusp form of weight k for SL2(Z) and λf(n) its n-th Fourier coefficient.In this paper,the exponential sum Xn 2X λf(n)e(αnβ) twisted by Fourier coefficients λf(n) is proved toh ave a main term of size |λf(q)|X3/4 when β = 1/2 and α is close to ±2√q,q ∈ Z,and is smaller otherwise for β 3/4.This is a manifestation of the resonance spectrum of automorphic forms for SL2(Z).     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Three-variable Mahler measures and special values of modular and Dirichlet L-series

In this paper we prove that the Mahler measures of the Laurent polynomials (x+x ^?1)(y+y ^?1)(z+z ^?1)+k ^1/2, (x+x ^?1)²(y+y ^?1)²(1+z)³ z ^?2?k, and x ⁴+y ⁴+z ⁴+1+k ^1/4 xyz, for various values of k, are of the form r ₁ L′(f,0)+r ₂ L′(χ,?1), where $r_{1},r_{2}\in \mathbb{Q}$ , f is a CM newform of weight 3, and χ is a quadratic character. Since it has been proved that these Mahler measures can also be expressed in terms of logarithms and ₅ F ₄-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.