A singular Sturm-Liouville equation under homogeneous boundary conditions

Given α>0 and f∈L²(0,1), we are interested in the equation

     

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let ?n∈C^∞(Rd?{0}) be a non-radial homogeneous distance function of degree n∈N satisfying ?n(tξ)=tn?n(ξ). For f∈S(Rd+1) and δ>0, we consider convolution operator associated with the smooth cone type multipliers defined by

     

Sobolev embeddings for Riesz potential spaces of variable exponents near 1 and Sobolev's exponent

Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of order α for functions f satisfying the Orlicz type condition

     

Weights whose biorthogonal polynomials admit a Rodrigues formula

Let α>0 and ψ(x)=xα. Let w be a non-negative integrable function on an interval I. Let Pn be a polynomial of degree n determined by the biorthogonality conditions

     

Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights

We completely describe those positive Borel measures μ in the unit disc D such that the Bergman space Ap(w)⊂Lq(μ), 0<p,q<∞, where w belongs to a large class W of rapidly decreasing weights which includes the exponential weights , α>0, and some double exponential type weights.As an application of that result, we characterize the boundedness and the compactness of Tg:Ap(w)→Aq(w), 0<p,q<∞, w∈W, where Tg is the integration operator

     

Singular integral operators for an unlimited contour

Let Γ be a closed or unclosed unlimited contour, a shift α(t) maps homeomorphically the contour Γ onto itself with preserving or reversing the direction on Γ and also satisfies the conditions for some natural number n?2, αn(t)≅t, and αj(t)?t for 1?j<n. In this work we study subalgebra Σ of algebra L(Lp(Γ,ρ)), which contains all operators of the form

     

Duality in Segal-Bargmann spaces

For α>0, the Bargmann projectionPα is the orthogonal projection from L²(γα) onto the holomorphic subspace , where γα is the standard Gaussian probability measure on Cn with variance (2α)⁻ⁿ. The space is classically known as the Segal-Bargmann space. We show that Pα extends to a bounded operator on Lp(γαp/2), and calculate the exact norm of this scaled Lp Bargmann projection. We use this to show that the dual space of the Lp-Segal-Bargmann space is an Lp^′ Segal-Bargmann space, but with the Gaussian measure scaled differently: (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms.     

Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk

Let D⊂R² be a disk, and let f∈C³. We assume that there is a∈R such that f(a)=0 and f^′(a)>0. In this article, we are concerned with the Neumann problem

     

Extension of Rotfel’d Theorem

Let f(t) be a non-negative concave function on the positive half-line. Given an arbitrary partitioned positive semi-definite matrix, we show that

     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

A class of Loewner chain preserving extension operators

We consider operators that extend locally univalent mappings of the unit disk Δ in C to locally biholomorphic mappings of the Euclidean unit ball B of Cn. For such an operator Φ, we seek conditions under which etΦ(e−tf(⋅,t)), t?0, is a Loewner chain on B whenever f(⋅,t), t?0, is a Loewner chain on Δ. We primarily study operators of the form , , where β∈[0,1/2] and is holomorphic, finding that, for ΦG,β to preserve Loewner chains, the maximum degree of terms appearing in the expansion of G is a function of β. Further applications involving Bloch mappings and radius of starlikeness are given, as are elementary results concerning extreme points and support points.     

Equivalence of some integrals and modulus of smoothness

Let f∈C(R). We are interested in lower and upper bounds of the integrals

     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V₀={p₁,p₂,p₃} be the set of vertices of SG and the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw=uw₁uw₂?uwn for any sequence w=(w₁,w₂,…,wn)∈n{1,2,3}. The union of the images of V₀ under these iterations is the set of nth stage vertices Vn of SG. Let F:Vn→R be any function. Given any numbers αw(w∈n{1,2,3}) with 0<|αw|<1, there exists a unique continuous extension of F, such that

f(uw(x))=αwf(x)+hw(x)     

Some new characterizations of Sobolev spaces

In this paper, we present some new characterizations of Sobolev spaces. Here is a typical result. Let g∈Lp(RN), 1<p<+∞; we prove that g∈W1,p(RN) if and only if

     

Global branch from the second eigenvalue for a semilinear Neumann problem in a ball

Let (n?3) be a ball, and let f∈C³. We are concerned with the Neumann problem

     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation

Let B be the unit ball in Cn with respect to an arbitrary norm and let f(z,t) be a g-Loewner chain such that e−tf(z,t)−z has a zero of order k+1 at z=0. In this paper, we obtain growth and covering theorems for . Moreover, we consider coefficient bounds and examples of mappings in .     

Young’s inequality and trace

Let φ be a positive linear functional on M_n(C) and f,g mutually conjugate in the sense of Young. In this note we show a necessary and sufficient condition for the inequality

     

Dynamical systems method (DSM) for selfadjoint operators

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av=f consists of solving the Cauchy problem , u(0)=u₀, where Φ is a suitable operator, and proving that (i) ∃u(t)∀t>0, (ii) ∃u(∞), and (iii) A(u(∞))=f. It is proved that if equation Av=f is solvable and u solves the problem , u(0)=u₀, where a>0 is a parameter and u₀ is arbitrary, then lima→0limt→∞u(t,a)=y, where y is the unique minimal-norm solution of the equation Av=f. Stable solution of the equation Av=f is constructed when the data are noisy, i.e., fδ is given in place of f, ‖fδ−f‖?δ. The case when a=a(t)>0, , a(t)↘0 as t→∞ is considered. It is proved that in this case limt→∞u(t)=y and if fδ is given in place of f, then limt→∞u(tδ)=y, where tδ is properly chosen.