Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants (Q2816966)

The authors extend results of \textit{E. B. Fabes} et al. [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 21, 147--157 (1975; Zbl 0336.31003); Indiana Univ. Math. J. 25, 159--170 (1976; Zbl 0306.46032)] and \textit{E. B. Fabes} and \textit{U. Neri} [Duke Math. J. 42, 725--734 (1975; Zbl 0331.35032)] from \(0 \leq \alpha < 1\) to \(-1 < \alpha <1\), as well as consider the scaling invariant version of the conditions they considered.NEWLINENEWLINEThe first spaces they consider, \(H^{\alpha, 2}\), as considered before, are defined by NEWLINE\[NEWLINE \sup_{(x_0,r) \in \mathbb R^{n+1}_+} r^{-(2 \alpha +n)} \int_{B(x_0, r)} \int_0^r | \nabla_{x,t}u(x, t)|^2 t \, \, dt \, dx < \infty, NEWLINE\]NEWLINE while the scaling invariant version is called \(\mathcal{H}^{\alpha,2}\) and is defined by NEWLINE\[NEWLINE \sup_{(x_0,r) \in \mathbb R^{n+1}_+} r^{-(2 \alpha +n)} \int_{B(x_0, r)} \int_0^r | \nabla_{x,t}u(x, t)|^2 t^{1 + 2 \alpha} \, \, dt \, dx < \infty. NEWLINE\]NEWLINE The trace at the boundary for the first condition is a function \(f\) in the Morrey-Campanato space \( \mathcal{L}_{2, n + 2 \alpha}\), which is defined by NEWLINE\[NEWLINE \| f \|_{ \mathcal{L}_{2, n + 2 \alpha}} := \left( \sup_{(x_0,r) \in \mathbb R^{n+1}_+} r^{-(2 \alpha +n)} \int_{B(x_0, r)} | f(x) - f_{B(x_0, r)} |^2 \, \, dx \right)^{1/2}, NEWLINE\]NEWLINE where \(f_{B(x_0, r)}\) is the integral average of \(f\) on \(B(x_0,r)\), NEWLINE\[NEWLINE f_{B(x_0, r)} = \frac{1}{| B(x_0, r)|} \int_{B(x_0,r)} f(y) \, \, dy. NEWLINE\]NEWLINE The trace at the boundary for the scaling invariant condition is \((-\Delta )^{\alpha/2} f\), for a function \(f\) in the Morrey-Campanato space \( \mathcal{L}_{2, n + 2 \alpha}\).NEWLINENEWLINEThe Morrey-Campanato spaces have found a new life in the study of nonlinear problems like the Navier-Stokes equation. The authors apply their result to give a result on well-posedness of a Navier-Stokes equation for \(\alpha \in (-1, 0]\) and on ill-posedness for \(\alpha \in (0,1)\) .NEWLINENEWLINEThe authors refer to the square Campanato spaces. \textit{H. Triebel} [Local function spaces, heat and Navier-Stokes equations. Zürich: European Mathematical Society (EMS) (2013; Zbl 1280.46002); Hybrid function spaces, heat and Navier-Stokes equations. Zürich: European Mathematical Society (EMS) (2015; Zbl 1330.46003)] refers to the Morrey spaces, but explains the distinction between Morrey's work and later work. I have used the hybrid, Morrey-Campanato spaces, in this review, although Campanato spaces may be more appropriate when one subtracts the average as they do. The Morrey-Campanato spaces were introduced for the study of the behavior of solutions of linear elliptic equations. In the study of linear problems, they were not as useful as possible because interpolation worked in only one direction. \textit{D. R. Adams} [Morrey spaces. Cham: Birkhäuser/Springer (2015; Zbl 1339.42001)] has returned to the topic and says that he has new results with some restrictions.NEWLINENEWLINEThe authors overlook one small point, which is obvious to experts, but should be mentioned. When they switch from Laplace's equation to the heat equation, they point out that the operator changes from \(e^{- t \sqrt{- \Delta}}\) to \(e^{t \Delta}\), which they do not define. The kernel in the definition of \(e^{t \Delta}\) switches from the Poisson kernel \(P\) (which they define) with Fourier transform \(e^{-t|\xi|}\) to the Gauss-Weierstrass kernel \(W\) (which they do not define) with Fourier transform \(e^{-t|\xi|^2}\), replacing NEWLINE\[NEWLINE P(x, t) = c_n t (|x|^2 + t^2)^{-\frac{n +1}{2}}, NEWLINE\]NEWLINE by NEWLINE\[NEWLINE W(x, t) = (2 \pi t)^{-n/2} e^{-|x|^2/4t}. NEWLINE\]NEWLINE The authors give a formula for the Poisson extension \(e^{- t \sqrt{- \Delta}}\). The reader can construct a similar formula for the Gauss-Weierstrass extension \(e^{t \Delta}\) by replacing \(P\) by \(W\) in their formula.