Smooth weak solutions by means of the wavelet transform (Q2890998)

The author studies the regularity of weak solutions of non-homogeneous, constant coefficient equations made of differential operators of pure order, \(Q=\sum_{|\alpha|=p}\partial^\alpha\) with \(p>1\), by using continuous wavelet transforms with respect to radially symmetric, admissible functions. If \([U(a,b)h](x)=1/a^{(n/2}h((x-b)/a)\), then \(h\in L^2({\mathbb R}^n)\) is said to be admissible if \(\int_G|(h,U(a,b)h)|^2\,d(a,b)<\infty\), where \(G\) is the affine, non unimodular group \(G=\{(a,b)\;|a>0, \text{and } b\in {\mathbb R}^n)\)\} with group operation \((a_1,b_1)=(a_1a_2,a_1b_2+b_1)\). The main result is in the following theorem: Suppose \(\Omega\) is a domain in \({\mathbb R}^n\) and \(u,f\in L^2({\mathbb R}^n)\). Suppose also that \(u\) is a weak solution of \(Qu=f\) in \(\Omega\). Then for \(h\) radially symmetric and admissible in \({\mathbb R}^n\), if \(f\) is of class \(C^\infty\) in a neighborhood of \(b_0\in\Omega\), then \(u\) is of class \(C^\infty\) in a neighborhood of \(b_0\in\Omega\).