A note on k-plane integral transforms

Let Π be a k-dimensional subspace of Rⁿ, n ? 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π^⊥. The k-plane transform of a measurable function ? in the direction Π at the point x″ is defined by $\text{L?(Π, x″) = ∝Π?(x′, x″) dx′}$. In this article certain a priori inequalities are established which show in particular that if $\text{? ? L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{)}$, $\text{1 ? p}\text{\$}\text{?}\text{n}\text{k}$, then ? is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces L^p(Rⁿ), p ? 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rⁿ are also obtained.     

Some combinatorial conjectures for Jack polynomials

We present several combinatorial conjectures related to the expansion of Jack polynomials in terms of power sums.     

Some vector-valued laplace transforms

Representation theorems for vector-valued Laplace transforms are discussed. Necessary and sufficient conditions are obtained in order that a function be the Laplace transform of a general vector measure and of a vector measure of finite variation, finiteq-variation or finiteq-semi-variation for 1<q≦∞.     

Majorants andL p norms

It is shown that ifG is an infinite compact abelian group, thenL ^p (G) has the upper majorant property only ifp is even orp=∞. The research for this paper was partially supported by National Research Council of Canada Operating Grant A-4822.     

Some geometric conjectures in harmonic function theory

We prove some results on the geometry of the level sets of harmonic functions, particularly regarding their ‘oscillation’ and ‘pinching’ properties. These results allow us to tackle three recent conjectures due to De Carli and Hudson (Bull London Math Soc 42:83–95, 2010). Our approach hinges on a combination of local constructions, methods from differential topology and global extension arguments.     

Some combinatorial conjectures for shifted Jack polynomials

We present several combinatorial conjectures related to Jack generalized binomial coefficients, or equivalently to shifted Jack polynomials. We prove these conjectures when the degree of these polynomials is 5.     

Some inequalities for Laplace transforms

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Some inequalities for characteristic functions, Theory Probab. Appl. 45 (2001) 711-713] about a lower bound of real-valued characteristic functions. In this paper, we investigate the counterparts for Laplace transforms of non-negative random variables. Surprisingly, the resulting inequalities hold true on the right half-line. Besides, we show some more inequalities by applying the convex/concave properties of the remainder in Taylor's expansion for the exponential function.     

Some theorems on integral transforms

A theorem is proved which establishes a connection among Laplace, Kantorovich-Lebedev, Meier, and the generalized Mehler-Focktransforms. Some improper integrals are calculated by using this theorem and its immediate generalization.     

Singular measures andl p Fourier transforms

LetG be the Cantor group or the circle group, and let Γ be the dual ofG. There exists a probability measure λ onG, singular with respect to Haar measure onG, such that for 1≦p<2 the inequality {fx115-1} holds for trigonometric polynomialsf. Partially supported by NSF Grant MCS 76-02267-A01.     

Some characterizations of orthant monotonic norms

In real n-space the orthant monotonic norms of Gries [5] can be given a new characterization similar to one for monotonic norms: a norm is orthant monotonic if and only if for every D=diag(δ₁,δ₂,…,δ_n)?0, the operator norm of D equals max δ_i. This gives an alternative proof to Gries's: a norm is orthant monotonic if and only if its dual norm is orthant monotonic. Also, it follows that the principal axis vectors are self-dual for orthant monotonic norms.     

Lieb-Thirring inequality for L p norms

In this paper, we obtain the Lieb-Thirring inequality for L _p -norms. The proof uses only the standard apparatus of the theory of orthogonal series.     

Some inverse Laplace transforms of exponential form

An algorithm for finding inverse Laplace transforms of exponential form for two classes of functions is established and used to derive a set of new formulas which are presented as a table of Laplace transform pairs. These formulas are useful in problems in fluid mechanics.