Hilbert’s inequality in several variables

The s -dimensional discrete and continuous Hilbert operators are introduced, extending the classical Hilbert inequality to several variables. Through an inductive argument, it is shown that in both cases the p -norm of the operator equals <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>π</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mi>csc</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mi>π</m:mi> <m:mi>p</m:mi> </m:mfrac> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>s</m:mi> </m:msup> </m:math> {(π(π/p))ˢ} , with the constant being optimal. The proofs rely on the factorization of the multi-dimensional kernel and the sharp one-dimensional inequality.