We prove the existence of steady periodic capillary water waves on flows with arbitrary vorticity distributions. They are symmetric two-dimensional waves whose profiles are monotone between crest and trough.

In this paper we generalize an interpolation result due to J.-O. Strömberg and A. Torchinsky to the case of one-sided Hardy spaces. This generalization is important in the study of the weak type (1,1) for lateral strongly singular operators. We shall need an atomic decomposition in which for every atom there exists another atom supported contiguously at its right. In order to obtain this decomposition we have developed a rather simple technique to break up an atom into a sum of others atoms.

Let $A=-(\nabla-i\vec{a})^2+V$ be a magnetic Schrödinger operator acting on$L$^{2}($R$^{$n$}),$n$≥1, where $\vec{a}=(a_1,\cdots,a_n)\in L^2_{\rm loc}$ and 0≤$V$∈$L$^{1}_{loc}. Following [1], we define, by means of the area integral function, a Hardy space$H$^{1}_{$A$}associated with$A$. We show that Riesz transforms (∂/∂$x$_{$k$}-$i$$a$_{$k$})$A$^{-1/2}associated with$A$, $k=1,\cdots,n$ , are bounded from the Hardy space$H$^{1}_{$A$}into$L$^{1}. By interpolation, the Riesz transforms are bounded on$L$^{$p$}for all 1<$p$≤2.

The α-modulation spaces$M$^{$s$,α}_{$p$,$q$}($R$^{$d$}), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ($x$,$D$) with symbol in the Hörmander class$S$^{$b$}_{ρ,0}extends to a bounded operator σ($x$,$D$):$M$^{$s$,α}_{$p$,$q$}($R$^{$d$})→$M$^{$s$-$b$,α}_{$p$,$q$}($R$^{$d$}) provided 0≤α≤ρ≤1, and 1<$p$,$q$<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class$S$^{$b$}_{1,0}maps the Besov space$B$^{$s$}_{$p$,$q$}($R$^{$d$}) into$B$^{$s$-$b$}_{$p$,$q$}($R$^{$d$}).

We give sharp estimates for the fractional maximal function in terms of Hausdorff capacity. At the same time we identify the real interpolation spaces between$L$_{1}and the Morrey space $\mathcal{L}^{1,\lambda}$ . The result can be viewed as an analogue of the Hardy–Littlewood maximal theorem for the fractional maximal function.