Generalized Lebesgue points for Sobolev functions

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space (X, d, μ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B(x, r) converge to f(x) when r converges to 0. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function f ∈ M ^s,p (X), 0 < s ≤ 1, 0 < p < 1, where X is a doubling metric measure space, has generalized Lebesgue points outside a set of \(\mathcal{H}^h\)-Hausdorff measure zero for a suitable gauge function h.     

Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with fractional Laplacian in the Schrödinger equation in R¹⁺¹. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schrödinger data in H^s1 and wave data in H^s2 ×H^s2?1 for 3/4?α < s₁ ≤ 0 and ?1/2 < s₂ < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α ≤ 1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s₁ = 0 and ?1/2 < s₂ < 1/2 by using the conservation law for the L² norm of u.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

On the positive definiteness of some functions related to the Schoenberg problem

For a broad class of functions f: [0,+∞) → ?, we prove that the function f(ρ ^λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ? 0 and α(E, ρ) is the Schoenberg constant.     

Composition limits and separating examples for some boolean function complexity measures

Block sensitivity (bs(f)), certificate complexity (C(f)) and fractional certificate complexity (C*(f)) are three fundamental combinatorial measures of complexity of a boolean function f. It has long been known that bs(f) ≤ C*(f) ≤ C(f) = O(bs(f)²). We provide an infinite family of examples for which C(f) grows quadratically in C*(f) (and also bs(f)) giving optimal separations between these measures. Previously the biggest separation known was \(C(f) = C*(f)^{\log _{4,5} 5}\). We also give a family of examples for which C*(f)= Ω (bs(f)^3/2).These examples are obtained by composing boolean functions in various ways. Here the composition fog of f with g is obtained by substituting for each variable of f a copy of g on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure s(f). The measures s(f), C(f) and C*(f) behave nicely under composition: they are submultiplicative (where measure m is submultiplicative if m(fog) ≤ m(f)m(g)) with equality holding under some fairly general conditions. The measure bs(f) is qualitatively different: it is not submultiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure m at function f, m ^lim(f) to be the limit as k grows of m(f ^(k))^1/k , where f ^(k) is the iterated composition of f with itself k-times. For any function f we show that bs ^lim(f) = (C*)^lim(f) and characterize s ^lim(f); (C*)^lim(f), and C ^lim(f) in terms of the largest eigenvalue of a certain set of 2×2 matrices associated with f.     

A note on the Schrödinger maximal function

It is shown that control of the Schrödinger maximal function sup_{0 <t<1} ?e^itΔf? for f ∈ H^s(Rⁿ) requires s ≥ n/2(n + 1).     

On the well-posedness of degenerate fractional differential equations in vector valued function spaces

We characterize completely the well-posedness on the vector-valued Hölder and Lebesgue spaces of the degenerate fractional differential equation D ^α (Mu)(t) = Au(t) + f(t), t ∈ ? by using vector-valued multiplier results in the spaces C ^γ (?;X) and L ^p (?;X), where A and M are closed linear operators defined on the Banach space X, 0 < γ < 1, 1 < p < ∞, the fractional derivative is understood in the sense of Caputo and α is positive.     

Estimates for solutions of linear and quasilinear systems in the nonautonomous case

By using the freezing method, we obtain upper and lower estimates for the higher and lower characteristic exponents, respectively, of homogeneous n-dimensional linear differential and difference systems with coefficient matrix A(t) satisfying the condition ||A(t)?A(s)|| ≤ δ|t ? s|^α, δ > 0, α > 0, t, s ≥ 0. We also prove analogs of these estimates for quasilinear differential and difference systems.     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.     

Fujita type conditions to heat equation with variable source

This paper studies heat equation with variable exponent u _t = Δu + u^p(x) + u ^q in ? ^N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p_? = inf p(x) ≤ p(x) ≤ sup p(x) = p₊. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p₊, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p₊ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p_? ≤ p₊ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p_? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p_? < 1 + \(\frac{2}{N}\) < p₊.     

Regularized traces of singular differential operators with canonical boundary conditions

A self-adjoint differential operator \(\mathbb{L}\) of order 2m is considered in L ₂[0,∞) with the classic boundary conditions \(y^{(k_1 )} (0) = y^{(k_2 )} (0) = y^{(k_3 )} (0) = \ldots = y^{(k_m )} (0) = 0\), where 0 ≤ k ₁ < k ₂ < ... < k _m ≤ 2m ? 1 and {k _s } _s=1 ^m ∪ {2m ? 1 ? k _s } _s=1 ^m = {0, 1, 2, ..., 2m ? 1}. The operator \(\mathbb{L}\) is perturbed by the operator of multiplication by a real measurable bounded function q(x) with a compact support: \(\mathbb{P}\) f(x) = q(x)f(x), f ∈ L ₂[0,∞). The regularized trace of the operator \(\mathbb{L} + \mathbb{P}\) is calculated.     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Unexpected spectral asymptotics for wave equations on certain compact spacetimes

We study the spectral asymptotics of wave equations on certain compact spacetimes, where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime S¹×S². For the Laplacian on S¹×S², theWeyl asymptotic law gives a growth rate O(s^3/2) for the eigenvalue counting function N(s) = #{λ_j: 0 ≤ λ_j ≤ s}. For the wave operator, there are two corresponding eigenvalue counting functions: N^±(s) = #{λ_j: 0 < ±λ_j ≤ s}, and they both have a growth rate of O(s²). More precisely, there is a leading term π²s²/4 and a correction term of as^3/2, where the constant a is different for N^±. These results are not robust in that if we include a speed of propagation constant to the wave operator, the result depends on number theoretic properties of the constant, and generalizations to S¹ × S^q are valid for q even but not q odd. We also examine some related examples.     

Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications

Let L _∞,s ¹ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that ?f/?x _i ∈ L _s (? ^{m) for each i = 1, ...,m} . New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥D ^α f∥_∞ of functions f ∈ L _∞,s ¹ (? ^m ). Stechkin’s problem on approximation of unbounded operators D ^α by bounded operators on the class of functions f ∈ L _∞,s ¹ (? ^m ) such that ∥?f∥ _s ≤ 1 and the problem of optimal recovery of the operator D ^α on elements from this class given with error δ are solved.     

Riemann Sums and Möbius

Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Möbius function has the property that it maps from \({ \cup _{0 < r < \infty }}{l^r}(s)\) to \({ \cap _{0 < r < \infty }}{l^r}(s)\), injectively. If 0 < r< 2 and ξ ∈ l^r (S), the series \({f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx} \) converges uniformly to an element of f_ξR₀, i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} {{f_\zeta }} (j/N) = 0,N = 1,2....\) If \({A_{\zeta \lambda }} = \lambda {\zeta _\lambda },{\zeta _\lambda }(1) = 1\), then ξ_λ is multiplicative. If \({f_{{\zeta _\lambda }}} \in {\Lambda _a}\), the space of α-Lipschitz continous functions, for some α > 0, and if χ is any Dirichlet character, then L(s, χ) ≠ 0, Res > 1 ? α. Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to f_ξ ∈ Λ_α, α < 1/2, ξ ∈ l^r (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies f_ξ ∈ Λ_α, α < 1/4, ξ ∈ l^r (S), 0 < r < 2. The question of whether R₀ ∩ Λ_α ≠ {0} for some positive α > 0 is open.     

Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with Jacobi weight ?^(α,β)(x) = (1 ? x)^α(1 + x)^β α ≥ β > ?1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L _q ^(α,β) , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained.     

Zeros of Jacobi polynomials Pn( α, β)$ P_{n}^{(\alpha ,\beta)} $, − 2 < α, β < − 1

The sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) is orthogonal on (??1,1) with respect to the weight function (1 ? x)^α(1 + x)^β provided α > ??1,β > ??1. When the parameters α and β lie in the narrow range ??2 < α, β < ??1, the sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) is quasi-orthogonal of order 2 with respect to the weight function (1 ? x)^α+?1(1 + x)^β+?1 and each polynomial of degree n,n ≥?2, in such a Jacobi sequence has n real zeros. We show that any sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) with ??2 < α, β < ??1, cannot be orthogonal with respect to any positive measure by proving that the zeros of Pn??1(α,β) do not interlace with the zeros of Pn(α,β) for any \(n \in \mathbb {N},\)n ≥?2, and any α,β lying in the range ??2 < α, β < ??1. We also investigate interlacing properties satisfied by the zeros of equal degree Jacobi polynomials Pn(α,β),Pn(α,β+?1) and Pn(α+?1,β+?1) where ??2 < α, β < ??1. Upper and lower bounds for the extreme zeros of quasi-orthogonal order 2 Jacobi polynomials Pn(α,β) with ??2 < α, β < ??1 are derived.     

On the Convergence of Solutions of Variational Problems with Implicit Pointwise Constraints in Variable Domains

Results on the convergence of minimizers and minimum values of integral and more general functionals J_s: W^1,p(Ω_s) → ? on the sets U_s(h_s) = {v ∈ W^1,p(Ω_s): h_s(v) ≤ 0 a.e. in Ω_s}, where p > 1, {Ω_s} is a sequence of domains contained in a bounded domain Ω of ?ⁿ (n > 2), and {h_s} is a sequence of functions on ?, are announced.     

Semi-Heavy Tails

In this paper, we study properties of functions and sequences with a semi-heavy tail, that is, functions and sequences of the form w(x) = e^?βxf(x), β > 0, resp., w_n = cⁿf_n, 0 < c < 1, where the function f(x), resp., the sequence (f_n), is regularly varying. Among others, we give a representation theorem and study convolution properties. The paper includes several examples and applications in probability theory.     

Behavior of the Fourier–Walsh coefficients of a corrected function

We prove that, given a sequence {ak}_k=1^∞ with a_k ↓ 0 and {ak}_k=1^∞ ? l₂, reals 0 < ε < 1 and p ∈ [1, 2], and f ∈ L^p(0, 1), we can find f ∈ L^p(0, 1) with mes{f ≠ f < ε whose nonzero Fourier–Walsh coefficients c_k(f) are such that |c_k(f)| = a_k for k ∈ spec(f).