Bilinear forms on homogeneous Sobolev spaces

In this paper we characterize the boundedness of the bilinear form defined by

$(f,g)\in {\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})\to \underset{\mathbb{R}}{\int }{(?\mathrm{\Delta })}^{s/2}(fg)(x){(?\mathrm{\Delta })}^{s/2}(b)(x)dx,$

in the product of homogeneous Sobolev spaces ${\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})$, $0<s<1/2$. We deduce a characterization of the space of pointwise multipliers from ${\dot{H}}^s(\mathbb{R})$ to its dual ${\dot{H}}^{?s}(\mathbb{R})$ in terms of trace measures.