A boundedness result for Marcinkiewicz integral operator

We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains {L}^{p} -bounded when its kernel satisfies only the sole integrability condition.

Growth conditions and regularity, an optimal local boundedness result

We prove local boundedness of local minimizers of scalar integral functionals [Formula: see text], [Formula: see text] where the integrand satisfies [Formula: see text]-growth of the form [Formula: see text] under the optimal relation [Formula: see text].

Feedback control for non-stationary 3D Navier–Stokes–Voigt equations

The goal of this article is to study the feedback control for non-stationary three-dimensional Navier–Stokes–Voigt equations. Based on the existence, uniqueness, and boundedness result of the weak solutions to the equations, we obtain the existence of solutions to the feedback control system. An existence result for an optimal control problem is also given. We illustrate our main result with an evolutionary hemivariational inequality.

The composition of linear canonical wavelet transforms on generalized function spaces

The main goal of this paper is to study the continuity of composition of linear canonical wavelet transform (LCWTs) on generalized test function spaces Lp,A, Gp,A and BA(R3). The boundedness result for composition of linear canonical wavelet transforms on Hps,A is given.

A sharp boundedness result for restricted maximal operators of Vilenkin–Fourier series on martingale Hardy spaces

The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space {H_{p}} to the Lebesgue space {L_{p}} for all {0<p\leq 1} . We also prove that the result is sharp in a particular sense.

Hilbert Transformation and Representation of the ax + b Group

In this paper we study the Hilbert transformations over L2() and L2() fromthe viewpoint of symmetry. For a linear operator over L2() commutative with the ax + b group, we show that the operator is of the form λI+ηH, where I and H are the identity operator and Hilbert transformation, respectively, and λ, η are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand–Naimark representation of the ax + b group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the ax + b group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle L2().