In this paper we study the Hilbert transformations over ${{L}^{2}}(\mathbb{R})$ and ${{L}^{2}}(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over ${{L}^{2}}(\mathbb{R})$ commutative with the $ax\,+\,b$ group, we show that the operator is of the form $\lambda I+\eta H$, where $I$ and $H$ are the identity operator and Hilbert transformation, respectively, and $\lambda ,\eta $ 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 ${{L}^{2}}(\mathbb{T})$.

We show that the discrete Hilbert transform and the discrete Kak–Hilbert transform are infinitesimal generators of one-parameter groups of operators in ${{\ell }^{2}}$.

In this paper we consider multipliers on the real Hardy space ${{H}_{2\pi }}$. It is known that the Marcinkiewicz and the Hörmander–Mihlin conditions are sufficient for the corresponding trigonometric multiplier to be bounded on $L_{2\pi }^{p},1<p<\infty$. We show among others that the Hörmander– Mihlin condition extends to ${{H}_{2\pi }}$ but the Marcinkiewicz condition does not.

In this paper we prove that if a weight w satisfies the condition, then the Lp(w) norm of a one-sided singular integral is bounded by the Lp(w) norm of the one-sided Hardy-Littlewood maximal function, for 1 < p < q < ∞.

We define Hardy spaces of conjugate systems of temperature functions on $\mathbb{R}_{+}^{n+1}$. We show that their boundary distributions are the same as the boundary distributions of the usual Hardy spaces of conjugate systems of harmonic functions.

We prove that the conjugate function operator is bounded in Lp(Г, wds), 1 < p < ∞, if and only if w ∈ Ap(Г), where Г is a quasiregular curve.

Suppose b(t) decreases to 0 on [1, ∞). Define the singular integral operator Cb at periodic f of period 1 in L1 (T),T = ( - 1 / 2, 1/2), by

Then, for a large class of b one has the rearrangement inequality

This inequality is used to construct a rearrangement invariant function space X corresponding to a given such space Y so that Cb maps X into Y.