Let X be a linear normed space, λ≥0, n∈ℕ. Let ????_λ⁽ⁿ⁾ be a set defined by
????_λ⁽ⁿ⁾ = {g: Xⁿ→ℂ |  |g(x^→)| ≤ M_g · e^{λΣ_k=1n‖x_k‖}, x^→∈Xⁿ},
where M_g is a constant depending on g. Moreover for all g∈????_λ⁽ⁿ⁾ we define
‖g‖ := sup_x^→∈Xⁿ {e^{-λΣ_k=1n‖x_k‖} · |g(x^→)|}.
In the paper norms of the d'Alembert and Lobaczevski difference operators in the ????_λⁿ spaces are calculated (their Pexider type generalizations are also considered). Moreover it is proved that if f: X→ℂ is a function such that A(f)∈????_λ⁽²⁾, where A is the d'Alembert difference operator, then f∈????_λ or A(f) = 0.

In this paper, the continuity of some multilinear integral operators on Morrey spaces are obtained. The operators contain singular integral operators, Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operators.

We formulate and generalize the technique of Jakubec established to derive congruences of Ankeny-Artin-Chowla type for a cyclic subfleld K of prime conductor p. Then we concentrate on the case of congruences modulo p³ and clear a significant technical hurdle which allows us to formulate Ankeny-Artin-Chowla congruences modulo p³ in a concise way.

The paper deals with an optimization problem in which minima of a finite collection of objective functions satisfy some unilateral constraints and are linked together by a certain subdifferential law. The governing relations are variational inequalities defined on a nonconvex feasible set. By reducing the problem to a variational inequality involving nonmonotone multivalued mapping defined over a nonnegative orthant, the existence of solutions is established under the assumption that constrained functions are positive homogeneous of degree at most one.

In this paper, the sharp estimates for some multilinear commutators related to certain sublinear integral operators are obtained. The operators include Littlewood-Paley operator and Marcinkiewicz operator. As application, we obtain the weighted L^p (p > 1) inequalities and LlogL type estimate for the multilinear commutators.

Report of Meeting. The Sixth Debrecen-Katowice Winter Seminar On Functional Equations and Inequalities February 1-4, 2006, Berekfürdő, Hungary.