BEST MONOTONE L_■-APPROXIMATIONS IN SEVERAL VARIABLES

Given a continuous function f defined on the unit cube of R~n and a convexfunction _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set ofbest L~(t)-approximations by monotone functions has exactly one elementft,which is also a continuous function.Moreover if the family of convexfunctions {_t}t>0 converges uniformly on compact sets to a function _0,then the best approximation f_t→f_0 uniformly,as t→0,where fo is thebest approximation of f within the Orlicz space L~(0) The best approxima-tions{f_t}are obtained as well as minimizing integrals or the Luxemburgnorm     

Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type

Suppose that ω(φ, ·) is the dyadic modulus of continuity of a compactly supported function φ in L ²(?⁺) satisfying a scaling equation with 2 ⁿ coefficients. Denote by α _φ the supremum for values of α > 0 such that the inequality ω(φ, 2^?j ) ≤ C2 ^?αj holds for all j ∈ ?. For the cases n = 3 and n = 4, we study the scaling functions φ generating multiresolution analyses in L ²(?₊) and the exact values of α _φ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in L ²(?₊) corresponding to the scaling function φ coincides with α _φ .     

On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary

We study the differentiability of very weak solutions v∈L¹(Ω) of ₀(v,L^?φ)=₀(f,φ) for all vanishing at the boundary whenever f is in L¹(Ω,δ), with δ=dist(x,∂Ω), and L^* is a linear second order elliptic operator with variable coefficients. We show that our results are optimal. We use symmetrization techniques to derive the regularity in Lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function of f.     

Closure properties of operators on the Ma-Minda type starlike and convex functions

A normalized univalent function f is called Ma-Minda starlike or convex if zf^′(z)/f(z)?φ(z) or 1+zf^″(z)/f^′(z)?φ(z) where φ is a convex univalent function with φ(0)=1. The class of Ma-Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and close-to-convex functions. Connections with various earlier works are made.     

Real analytic expansion of spectral projections and extension of the Hecke-Bochner identity

In this article, we review the Weyl correspondence of bigraded spherical harmonics and use it to extend the Hecke-Bochner identities for the spectral projections f × φ _k ^n?1 for function f ∈ L ^p (? ⁿ ) with 1 ≤ p ≤ ∞. We prove that spheres are sets of injectivity for the twisted spherical means with real analytic weight. Then, we derive a real analytic expansion for the spectral projections f × φ _k ^n?1 for function f ∈ L ²(? ⁿ ). Using this expansion we deduce that a complex cone can be a set of injectivity for the twisted spherical means.     

Toeplitz operators in several complex variables

Let S be the unit sphere in Cⁿ. We investigate the properties of Toeplitz operators on S, i.e., operators of the form Tφf = P(φf) where φ?L^∞(S) and P denotes the projection of L²(S) onto H²(S). The aim of this paper is to determine how far the extensive one-variable theory remains valid in higher dimensions. We establish the spectral inclusion theorem, that the spectrum of T_φ contains the essential range of φ, and obtain a characterization of the Toeplitz operators among operators on H²(S) by an operator equation. Particular attention is paid to the case where φ ? H^∞(S) + C(S) where C(S) denotes the algebra of continuous functions on S. Finally we describe a class of Toeplitz operators useful for providing counterexamples—in particular, Widom's theorem on the connectedness of the spectrum fails when n > 1.     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

Boundary behavior of Orlicz-Sobolev classes

It is proved that homeomorphisms of the Orlicz-Sobolev class W _loc ^{1, φ} can be continuously extended to the boundaries of some domains if the function φ defining this class satisfies a Carderón-type condition and the outer dilatation K _f of the mapping f satisfies the divergence condition for integrals of special form. In particular, the result holds for homeomorphisms of the Sobolev classes W _loc ^1,1 with K _f ∈ L _loc ^q for q > n ? 1.     

On the mean oscillation of the Hessian of solutions to the Monge-Ampère equation

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation detD²u=f(x) with zero boundary values, where f(x) is a non-Dini continuous function. If the modulus of continuity of f(x) is φ(r) such that limr→0φ(r)log(1/r)=0, then D²u∈VMO.     

Finite range decompositions of positive-definite functions

We give sufficient conditions for a positive-definite function to admit decomposition into a sum of positive-definite functions which are compactly supported within disks of increasing diameters Ln. More generally we consider positive-definite bilinear forms f→v(f,f) defined on . We say v has a finite range decomposition if v can be written as a sum v=∑Gn of positive-definite bilinear forms Gn such that Gn(f,g)=0 when the supports of the test functions f,g are separated by a distance greater or equal to Ln. We prove that such decompositions exist when v is dual to a bilinear form φ→∫²|Bφ| where B is a vector valued partial differential operator satisfying some regularity conditions.     

Regularity property of Donsker's delta function

LetL be the space of rapidly decreasing smooth functions on ? andL ^* its dual space. Let (L ²)⁺ and (L ²)^? be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL ^* with standard Gaussian measure. The Donsker delta functionδ(B(t)?x) is in (L ²)^? and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH _n is the Hermite polynomial of degreen. It is shown that forφ in (L ²)⁺,g _t,φ(x)≡〈δ(B(t)?x), φ〉 is inL and the linear map takingφ intog _t,φ is continuous from (L ²)⁺ intoL. This implies that forf inL ^* is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ _n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL ^*, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where? _s ^* is the adjoint of \(\dot B(s)\) -differentiation operator? _s .     

Linear transformations of monotone functions on the discrete cube

For a function f:{0,1}ⁿ→R and an invertible linear transformation L∈GL_n(2), we consider the function Lf:{0,1}ⁿ→R defined by Lf(x)=f(Lx). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I(Lf)≥I(f), where I(f) is the total influence of f. Second, we conjecture that if both f and L(f) are monotone, then f=L(f) (up to a permutation of the coordinates). We prove the second conjecture in the case where L is upper triangular.     

Spanning forests and the golden ratio

For a graph G, let f_ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_ij's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F=(f_ij)_n×n/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^-1=I+L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to φ^-1 or to 1-φ^-1, where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts” and “golden extroverts,” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.     

Best φ-approximation by n-convex functions

A n-convex function defined on a bounded open interval J ₀ n ≥2 is the (n?l)-st indefinite integral of a nondecreasing function. This fact and the simple structure of the latter enable to obtain concrete results about a n-convex best φ approximation g to a function f ? L _φ on J ₀, where φ: [0, ∞) → [0, ∞) is a convex function that generaJizes the p^th -power functions, 1 ≤ p < ∞. It is shown that g may also be a best generalized spline φ approximation to the restriction of f on the maximal subintervals of J₀ where g is a generalized spline. This is the situation in some cases, among which the L_p -approximation is includedp ≥ 1. For n = 2 it is proven that g is a polynomial of best φ-approximation to f ? L _φ on any maximal interval where g is a polynomial. If f is in addition continuous, then this fact implies the uniqueness of g Under the same assumption, it is shown that the best 3-convex L ₁-approximation is also unique whenever its derivative is bounded.     

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Let $\mathcal{X}$ be a metric space with doubling measure and L a nonnegative self-adjoint operator in $L^{2}(\mathcal{X})$ satisfying the Davies–Gaffney estimates. Let $\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that φ(x,?) is an Orlicz function, $\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2?I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space $H_{\varphi,L}(\mathcal{X})$ , by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space $\mathrm{BMO}_{\varphi,L}(\mathcal{X})$ , which is further proved to be the dual space of $H_{\varphi,L}(\mathcal{X})$ and hence whose φ-Carleson measure characterization is deduced. Characterizations of $H_{\varphi,L}(\mathcal{X})$ , including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize $H_{\varphi,L}(\mathcal{X})$ in terms of the Littlewood–Paley $g^{\ast}_{\lambda}$ -function $g^{\ast}_{\lambda,L}$ and establish a Hörmander-type spectral multiplier theorem for L on $H_{\varphi,L}(\mathcal{X})$ . Moreover, for the Musielak–Orlicz–Hardy space H _φ,L (? ⁿ ) associated with the Schrödinger operator L:=?Δ+V, where $0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ , the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ?L ^?1/2 is bounded from H _φ,L (? ⁿ ) to the Musielak–Orlicz space L ^φ (? ⁿ ) when i(φ)∈(0,1], and from H _φ,L (? ⁿ ) to the Musielak–Orlicz–Hardy space H _φ (? ⁿ ) when $i(\varphi)\in(\frac{n}{n+1},1]$ , where i(φ) denotes the uniformly critical lower type index of φ.     

Approximation by antiderivatives

Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ ^(?n)} ofn-fold antiderivatives (i.e., we haveφ ⁽⁰⁾(z)∶=φ(z) andφ ^(?n)(z)=dφ ^(?n?1)(z)/dz for alln ∈ N₀ and z ∈ Ω) such that the following properties hold:

1. For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n _k} such that {φ^?nk} converges tof uniformly onB.
2. For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m _k} such that {φ^?mk} converges tof compactly onU.
3. For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p _k} such that {φ ^(-Pk)} converges tof almost everywhere onE.

     

Closable Multipliers

Let (X, μ) and (Y, ν) be standard measure spaces. A function \({\varphi\in L^\infty(X\times Y,\mu\times\nu)}\) is called a (measurable) Schur multiplier if the map S _φ , defined on the space of Hilbert-Schmidt operators from L ₂(X, μ) to L ₂(Y, ν) by multiplying their integral kernels by φ, is bounded in the operator norm. The paper studies measurable functions φ for which S _φ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if φ is of Toeplitz type, that is, if φ(x, y) = f(x ? y), \({x,y\in G}\), where G is a locally compact abelian group, then the closability of φ is related to the local inclusion of f in the Fourier algebra A(G) of G. If φ is a divided difference, that is, a function of the form (f(x) ? f(y))/(x ? y), then its closability is related to the “operator smoothness” of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.     

A Class of Operators on Lh 2

Let D be the open unit disc in ? and let L_h ² be the space of quadratic integrable harmonic functions defined on D. Let \(\varphi: {\bar D}\rightarrow {\rm C}\) be a function in L^∞(D) with the property that φ(b) = lim_x→b,x?Dφ(x) for all b ? ?D. Define the operator C_φ in L_h ² as follows: C_φf = Q(φ·f),f ? L_h ², where Q is the orthogonal projection from L² (D) on L_h ². The following results are proved. If φ¦_?D ≡ 0, then C_φ is a compact linear operator and if φ¦_?D vanishes nowhere, then C_φ is a Fredholm operator.     

An inversion formula for a class of integral transforms,II

In this note we give a procedure for inverting the integral transform f(x) = ∫₀^∞k(xt) φ(t) dt, where the functions f(x) and k(x) are known and φ(x) is to be found. The inversion is accomplished in two steps: by first defining a transforming function, which is an integral, followed by the application of an infinite order differential operator.