Mathematics

A real valued function f defined on a real open interval I is called Φ -monotone if, for all x,y∈I with x≤y it satisfies f(x)≤f(y)+Φ(y−x), where Φ:[0,ℓ(I)[→ R + is a given nonnegative error function, where ℓ(I) denotes the length of the interval I . If f and −f are simultaneously Φ -monotone, then f is said to be a Φ -Hölder function. In the main results of the paper, using the notions of upper and lower interpolations, we establish a characterization for both classes of functions. This allows one to construct Φ -monotone and Φ -Hölder functions from elementary ones, which could be termed the building blocks for those classes. In the second part, we deduce Ostrowski- and Hermite--Hadamard-type inequalities from the Φ -monotonicity and Φ -Hölder properties, and then we verify the sharpness of these implications. We also establish implications in the reversed direction.

Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how certain sums of such polynomials with a fixed total degree relate to some standard polynomials.

In the abstract of [1] we read: "We obtain so far unproved properties of a ratio involving a classof Hermite and parabolic cylinder functions." However, we explain how some of the main results in that paper were already proved in [2], namely the `universal bounds'. An error in reference [2] was discussed in [1] which does not affect the proof given there for those `universal bounds'; we fix this erratum easily. We end this note proposing a conjecture regarding the best possible upper bound for a certain ratio of parabolic cylinder functions.

We answer a question of Banakh, Jabłońska and Jabłoński by showing that for d≥2 there exists a compact set K⊆ R d such that the projection of K onto each hyperplane is of non-empty interior, but K+K is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a K in the unit cube with full projections, that is, such that the projections of K agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for ℓ -fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.

We show that the X-ray transform restricted along the moment curve possesses extremizers and that L p -normalized extremizing sequences are pre-compact modulo symmetry.

In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.

Let μ be a Radon measure on R d . We define and study conical energies E μ,p (x,V,α) , which quantify the portion of μ lying in the cone with vertex x∈ R d , direction V∈G(d,d−n) , and aperture α∈(0,1) . We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that μ has polynomial growth, we give a sufficient condition for L 2 (μ) -boundedness of singular integral operators with smooth odd kernels of convolution type.

We solve the connection problem of a certain system of linear q -difference equations recently introduced by K. Park. The result contains the connection formulas of the q -Lauricella hypergeometric function ? D and those of the q -generalized hypergeometric function N+1 ? N as special cases. Our result gives a simultaneous extension of them.

The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various kind of driving paths. We also extend the notion of driving rough path. We also introduce as an example a new family of branched rough paths, called aromatic rough paths modeled after aromatic Butcher series.

We consider the most general class of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of an arbitrary order whose solutions and right-hand sides belong to appropriate Sobolev spaces. For parameter-dependent problems from this class, we prove a constructive criterion for their solutions to be continuous in the Sobolev space with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem.