Conjugate temperatures (Q1191738)

The authors study pairs \((u,v)\) of temperatures on \(\mathbb{R}\times]0,\infty[\) that satisfy the equations \[ D_ x u(x,t)=-iD_ t^{1/2} v(x,t) \qquad\text{and}\qquad iD_ t^{1/2} u(x,t)=D_ x v(x,t), \] which are analogous to the Cauchy-Riemann equations; here \(D_ t^{1/2}\) is a Weyl fractional derivative. In particular, they define an operator \(\tilde S\) on \(L^ 1(\mathbb{R},dx/(1+x^ 2))\) such that, if \(k\) is the Gauss-Weierstrass kernel, then \((u,v)=(g*k,\tilde Sg)\) satisfies the above equations, and \(\tilde S g(\cdot,0+)\) is the Hilbert transform of \(g\).