By Juan M. Delgado Sanchez, Tomas Dominguez Benavides
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Extra info for Advanced Course Of Mathematical Analysis III: Proceedings of the Third International School La Rabida, Spain, 3 - 7 September 2007
Lp3 (µ) Given N ∈ N, integrating over v ∈ [−N, N ], 2N TK (f, g) p3 Lp3 (µ) p3 N ≤ dm(v). (Rv−u f )(Rv+u g)K(u)du −N R Lp3 (µ) May 6, 2008 15:45 36 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 O. Blasco Therefore 2N TK (f, g) p3 Lp3 (µ) N A −N −A = p3 N ≤ −N Ω R p3 dv Rv−u f (w)Rv+u g(w)K(u)du Ω dµ(w) BK (Ru f (w)χ[−A−N,A+N ] , Ru g(w)χ[−A−N,A+N ])(v) = Ω dµ(w)dv Rv−u f (w)Rv+u g(w)K(u)du p3 dv R × dµ(w) = BK (Ru f (w)χ[−A−N,A+N ], Ru g(w)χ[−A−N,A+N ] ) Ω ≤ Np1 ,p2 (BK )p3 Ru f (w)χ[−A−N,A+N ] Ω × = Np1 ,p2 (BK ) p3 Ru g(w)χ[−A−N,A+N ] Ω Ru f p1 Lp1 (µ) = Np1 ,p2 (BK ) Ru g −(A+N ) p2 Lp2 (R) p3 /p2 dµ(w) du p2 Lp2 (µ) du p3 /p1 A+N f −(A+N ) p1 Lp1 (µ) p2 Lp2 (µ) g −(A+N ) ≤ Np1 ,p2 (BK )p3 (2(A + N )) f du p3 /p2 A+N × p3 /p1 dµ(w) p3 /p2 A+N × dµ(w) p3 /p1 A+N −(A+N ) p3 p1 Lp1 (R) p3 Lp3 (R) p3 Lp1 (µ) g du p3 Lp2 (µ) .
Amer. Math. Soc. 70 (1964), 566–569. 9. N. Nikolskii, Selected problems of weighted approximation and spectral analysis, Trudy Math. Inst. Steklov 120 (1974). English translation in Proc. Steklov Math. Inst. 120 (1974). 10. H. J. Landau, A sparse sequence of exponentials closed on large sets, Bull. Amer. Math. Soc. 70 (1964), 566–569. 11. A. Olevskii, Completeness in L2 (R) of almost integer translates, C. R. Acad. Sci. Paris 324(1) (1997), 987–991. 12. A. Olevskii and A. , J. Fourier Anal. Appl.
E. Berkson, O. Blasco, M. Carro and A. Gillespie, Discretization and transference of bisublinear maximal operators, J. Fourier Anal. and Appl. 12 (2006), 447–481. 3. E. Berkson, O. Blasco, M. Carro and A. Gillespie, Discretization versus transference for bilinear operators, (2008), to appear. 4. O. Blasco, M. Carro and A. Gillespie, Bilinear Hilbert transform on measure spaces, J. Fourier Anal. and Appl. 11 (2005), 459–470. 5. O. Blasco and F. Villarroya, Transference of bilinear multipliers on Lorentz spaces, Illinois J.
Advanced Course Of Mathematical Analysis III: Proceedings of the Third International School La Rabida, Spain, 3 - 7 September 2007 by Juan M. Delgado Sanchez, Tomas Dominguez Benavides