Generalized conjugate partial sums of Fourier series are used to find jumps of functions. The rate of convergence is studied and sharp results are obtained.

Proposition 4.1(i) of [1] is incorrect, i.e. the sequence of Cesàro-sections ${\sigma }_{n}x$ of a sequence x in a translation invariant BK-space is not necessarily bounded. Theorem 4.2(ii) of [1] and the proof of Proposition 4.3 of [1] are corrected. All other statements of [1], including Proposition 4.3 itself, are correct.

We consider the maximal function $\left|\right|\left(S^{a}f\right)\left[x\right]{\left|\right|}_{L^{\infty }[-1,1]}$ where $\left(S^{a}f\right){\left(t\right)}^{\wedge }\left(\xi \right)=e^{{it\left|\xi \right|}^a}f̂\left(\xi \right)$ and 0 < a < 1. We prove the global estimate $\left|\right|S^{a}f{\left|\right|}_{L²(ℝ,L^{\infty }[-1,1])}{\le C\left|\right|f\left|\right|}_{H^s\left(ℝ\right)}$, s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

We prove the $L^p$ boundedness of certain nonconvolutional oscillatory integral operators and give explicit description of their extended domains. The class of phase functions considered here includes the function ${\left|x\right|}^{\alpha }{\left|y\right|}^{\beta }$. Sharp boundedness results are obtained in terms of α, β, and rate of decay of the kernel at infinity.

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(\alpha ₁,...,{\alpha }_N)$ for some $0<\alpha ₁,...,{\alpha }_N<1$ and its marginal functions satisfy $f(·,x₂,...,x_N)\in Lip\beta ₁,...,f(x₁,...,x_{N-1},·)\in Lip{\beta }_N$ for some $0<\beta ₁,...,{\beta }_N<1$ uniformly in the indicated variables $x_l$, 1 ≤ l ≤ N, then $f{̃}^{(\eta ₁,...,{\eta }_N)}\in Lip(\alpha ₁,...,{\alpha }_N)$ for each choice of $(\eta ₁,...,{\eta }_N)$ with ${\eta }_l=0$ or 1 for 1 ≤ l ≤ N.