On the normal solvability of pseudodifferential operators in the half space with piecewise continuous symbol
✍ Scribed by Dieter Heunemann; Jürgen Leiterer
- Publisher
- John Wiley and Sons
- Year
- 1980
- Tongue
- English
- Weight
- 270 KB
- Volume
- 95
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Let R"+ ={([,, . . . , tn)€R": CnsO}. We denote by P the orthogonal projection from L2(Rn) onto L,(R:). By P is denoted the FOURIER transformation in L3( Rn) :
Pi([) = J f ( z ) e-z(z*t)dz .
Rn
We consider the pseudodifferential operator A = PF-IuF acting in the space L,(R'L,), where the symbol a is a bounded measurable complex function on R" (see, for example [ 5 ] ) . As in [ 5 ] we assume that the symbol is positively homogeneous: a(tC) = a ( [ ) for t S O . However, in distinction to [ 5 ] , we admit that the symbol is only piecewise continuous (in the sense defined below).
We give a necessary and sufficient condition that the operator A is normally solvable in L,(R",). Here normal solvability means that the image Im A is closed, whereas the numbers dim Ker A and codim A can be infinite. It turns out t h a t