$L^p$-boundedness for pseudodifferential operators with non-smooth symbols and applications

Garello, Gianluca ; Morando, Alessandro

Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 2, pp. 461-503.

Abstract (VO)

Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of $L^{p}$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0 \rho \leq 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.

Riassunto

Utilizzando una formulazione generalizzata della caratterizzazione per corone diadiche degli spazi di Sobolev, nel presente lavoro si dimostra la continuità $L^{p}$ per operatori pseudodifferenziali il cui simbolo a(x,ξ) non è infinitamente differenziabile rispetto alla variabile x, mentre le sue derivate rispetto alla variabile ξ decadono con ordine ρ, con $0 \rho \leq 1$. Viene poi provata una proprietà di algebra per una classe di spazi di Sobolev pesati, che ben si applica allo studio della regolarità delle soluzioni di equazioni semi lineari multi-quasi-ellittiche.

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