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Vector-valued Hilbert transforms along curves. (English) Zbl 1337.43003
Summary: We show that Hilbert transforms along some curves are bounded on \(L^{p}({\mathbb{R}}^{n};X)\) for some \(1< p< \infty\) and some UMD spaces \(X\). In particular, we prove that Hilbert transforms along some curves are completely \(L^{p}\)-bounded in the terminology from operator space theory. Moreover, we obtain the \(L^{p}(\mathbb{R}^{n};X)\)-boundedness of anisotropic singular integrals by using the “method of rotations” of Calderón-Zygmund. All these results extend preexisting related ones.

MSC:
43A32 Other transforms and operators of Fourier type
46B99 Normed linear spaces and Banach spaces; Banach lattices
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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