Volume 5, Issue 2

It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

An asymptotic matrix solution is formulated for a class of mixed-type linear vector equations with a single variable deviation which is small at infinity. This matrix solution describes the asymptotic behavior of all exponentially bounded solutions. A sufficient condition is obtained for there to be no other solutions.

Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

Some multidimensional versions of a characteristic problem for second-order degenerating hyperbolic equations are considered. Using the technique of functional spaces with a negative norm, the correctness of these problems in the Sobolev weighted spaces are proved.

It is proved that for any given sequence (σ n , n ∈ ℕ) = Γ 0 ⊂ Γ, where Γ is the set of all directions in (i.e., pairs of orthogonal straight lines) there exists a locally integrable function f on such that: (1) for almost all directions σ ∈ Γ\Γ 0 the integral ∫ f is differentiable with respect to the family B 2σ of open rectangles with sides parallel to the straight lines from σ; (2) for every direction σ n ∈ Γ 0 the upper derivative of ∫ f with respect to B 2σ n equals +∞; (3) for every direction σ ∈ Γ the upper derivative of ∫ | f | with respect to B 2σ equals +∞.

Conditions on weights 𝑢 (·), υ (·) are given so that a classical operator T sends the weighted Lorentz space L rs ( υd𝑥 ) into L pq ( υd𝑥 ). Here T is either a fractional maximal operator M α or a fractional integral operator I α or a Calderón–Zygmund operator. A characterization of this boundedness is obtained for M α and I α when the weights have some usual properties and max( r, s ) ≤ min( p, q ).