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Silvia I. Hartzstein, Beatriz E. Viviani  JDA ?FIEJE B JDA EJACH= =@ @AHEL=JELA FAH=JHI B BK?JE= H@AH

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Silvia I. Hartzstein, Beatriz E. Viviani

On the composition of the integral and derivative operators of functional order

Comment.Math.Univ.Carolinae 44,1 (2003) 99-120.

Abstract: The Integral, Iφ, and Derivative, Dφ, operators of order φ, with φ a function of positive lower type and upper type less than 1, were defined in [HV2]

in the setting of spaces of homogeneous-type. These definitions generalize those of the fractional integral and derivative operators of orderα, whereφ(t) =tα, given in [GSV].

In this work we show that the composition Tφ = Dφ◦Iφ is a singular integral operator. This result in addition with the results obtained in [HV2] of boundedness of Iφ and Dφ or the T1-theorems proved in [HV1] yield the fact that Tφ is a Calder´on-Zygmund operator bounded on the generalized Besov, ˙Bpψ,q, 1≤p, q <∞, and Triebel-Lizorkin spaces, ˙Fpψ,q, 1 < p, q < ∞, of order ψ = ψ1/ψ2, where ψ1 and ψ2 are two quasi-increasing functions of adequate upper types s1 and s2, respectively.

Keywords: fractional integral operators, fractional derivative operators, spaces of homogeneous type, Besov spaces, Triebel-Lizorkin spaces

AMS Subject Classification: 26A33

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