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Singular integrals associated to the Laplacian on the affine group \(ax+b\). (English) Zbl 0776.43003

Let \(G\) be the affine group over the real line. Let \(X\) and \(Y\) be right-invariant differential operators induced by the elements of the basis for the Lie algebra of \(G\); \(L\) denotes the closure of \(-X^2 - Y^2\) on \(L^2(G)\). Let \(Z_1\), \(Z_2\) be any nonzero operators in the span of \(X\) and \(Y\). The authors establish the following results: If \(p\in ]1,+\infty[\), \(Z_1 L^{-1}Z_2\) defines a bounded operator from \(L^ p(G)\) into \(L^p(G)\) and a bounded operator from \(L^1(G)\) into \((L^1(G)\),weak); if \(p\in [1,+\infty[\), the operators \(Z_1 Z_2 L^{-1}\) and \(L^{-1} Z_1 Z_2\) are not of weak type \((p,p)\). In the proofs the kernels of convolution operators are split into their local parts and their parts at infinity.

MSC:

43A80 Analysis on other specific Lie groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
44A10 Laplace transform
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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