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Appl. 7,273 (1987). 6. F. Chiarenza, M. Frasca, P. Longo, Ricerche Mat. 40, 149 (1991). 7. R. Coifman, R. Rochberg, G. Weiss,Ann. of Math. 103,611 (1976). 8. E. B. Fabes, N. RiviBre, Studia Math. 27, 19 (1966). 9. J. Garcia-Cuerva, J . L. Rubio De Francia, Weighted Norm Inequalities and Related Topics: North-Holand Math. Studies, Vol. 116, North-Holand, Amsterdam, 1985. 10. F. John, L. Nirenberg,Comm. Pure Appl. Math. 14,415 (1961). 11. W. Jones, Indiana Uniu. Math. J . 29,41 (1980). 12. E. Nakai, Math.

However, its Mellin transform k * ( s ) is a quotient of products of Gamma functions] and therefore has a very simple asymptotic behavior a t infinity, powerexponential decay. We introduce now a space of functions M;;(L,) based on the power-exponential asymptotic behavior of k * ( s ) on %s = l / 2 . For simplicity we assume p = 2. 37 This space turns out to be very convenient in studying convolution G-transform (56). 1. Let 2 signc+signy 2 0. If the norm o f f in M;:(L2) is defined through the norm of f*(s), then M;;(Lz) is a Banach space.

S ~ (15) (16) + for [<'I 141 # 0, has a unique solution in the space M of all stable solutions of (15) for arbitrary hj. The problem (13) - (14) is said to be elliptic if g A ( < , 4 ) # 0, for + Iql # 0, 0 the problem satisfies the condition Shapiro - Lopatinski. 1. Let p , C be in R such that C, 5 C, 1 5 p < 00, and the problem (13) - (14) is elliptic. Then, the following statements hold true. ( i ) If q E Q \ (0) then U has the inverse operator U-' is a bounded linear operator f r o m H E , ~ , , ( RRn-') ~ + , t o H e , p , , ( R ~ )not , depending o n p , C, (aa) If q = 0 , there exists a bounded linear operator R f r o m H ~ , p ~ q ( R T , to He,P,q(RT),not depending o n p , C such that UR=IfT, where, I is the identity operator o n He,p(R"+RIW"-l) T is a bounded linear operator f r o m He,p,q(RT,Rn-') to H ~ + I , ~ , , ( RRn-').