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Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants. (English) Zbl 1366.46022

The authors extend results of 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 E. B. Fabes and 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.
The first spaces they consider, \(H^{\alpha, 2}\), as considered before, are defined by \[ \sup_{(x_0,r) \in \mathbb R^{n+1}_+} r^{-(2 \alpha +n)} \int_{B(x_0, r)} \int_0^r | \nabla_{x,t}u(x, t)|^2 t \, \, dt \, dx < \infty, \] while the scaling invariant version is called \(\mathcal{H}^{\alpha,2}\) and is defined by \[ \sup_{(x_0,r) \in \mathbb R^{n+1}_+} r^{-(2 \alpha +n)} \int_{B(x_0, r)} \int_0^r | \nabla_{x,t}u(x, t)|^2 t^{1 + 2 \alpha} \, \, dt \, dx < \infty. \] 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 \[ \| f \|_{ \mathcal{L}_{2, n + 2 \alpha}} := \left( \sup_{(x_0,r) \in \mathbb R^{n+1}_+} r^{-(2 \alpha +n)} \int_{B(x_0, r)} | f(x) - f_{B(x_0, r)} |^2 \, \, dx \right)^{1/2}, \] where \(f_{B(x_0, r)}\) is the integral average of \(f\) on \(B(x_0,r)\), \[ f_{B(x_0, r)} = \frac{1}{| B(x_0, r)|} \int_{B(x_0,r)} f(y) \, \, dy. \] The trace at the boundary for the scaling invariant condition is \((-\Delta )^{\alpha/2} f\), for a function \(f\) in the Morrey-Campanato space \( \mathcal{L}_{2, n + 2 \alpha}\).
The 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 \(\alpha \in (0,1)\) .
The authors refer to the square Campanato spaces. 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. 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.
The 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 \[ P(x, t) = c_n t (|x|^2 + t^2)^{-\frac{n +1}{2}}, \] by \[ W(x, t) = (2 \pi t)^{-n/2} e^{-|x|^2/4t}. \] 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.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
42B37 Harmonic analysis and PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
35Q30 Navier-Stokes equations
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References:

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