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by: Kaylin Wehner
Kaylin Wehner
GPA 3.55


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About this Document

Class Notes
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This 3 page Class Notes was uploaded by Kaylin Wehner on Friday September 4, 2015. The Class Notes belongs to MATH 1 at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/177817/math-1-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.


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Date Created: 09/04/15
Some open problems in mathematics Christoph Thiele July 27 2008 These are some of my favorite open problems in mathematics 1 Tri linear Hilbert transform Let 04 be an irrational number For compactly supported smooth functions f1f2f3f4 on R de ne Af1f2fsf4 Rpv Rf196 7 mm mm 7 at from Here the principal value is de ned as dt dt pvilim quot7 R t H0 rupee t Prove or disprove here HfH4 denotes the L4 norm Conjecture 1 There is a constant 0 independent of f1 f4 such that lAf1f2fsf4l S OHflH4Hf2H4Hf3H4Hf4H4 To trace some background information start with 2 Nonlinear Carleson theorem Let V be a function in L2R Then for h 2 07 by elementary methods7 the ordinary differential equation f W kf 1 has a two dimensional space of classical solutions with absolutely continuous derivatives satisfying the ode almost everywhere on R Prove or disprove Conjecture 2 For V E L2R there is a set of measure zero in R4r such that for h E R4r not in this set all solutions to the above ode are bounded Lm functions To trace some background information start with 3 TensorParaproduct Let b be a non zero Schwarz function o E SR7 think of it as a smooth approximation to the characteristic function of 711 De ne 45W 2 k 2 k9 and the smoothing operators at scale 2k Pka f bk This is a smooth version of the standard martingale average operator that averages on dyadic intervals of length 2k7 and the problem below is equally little understood in the case of the standard martingale operator De ne the difference operator Qk Pk 7 Pk1 Turning to functions in two variables x and y7 de ne PM and QM the corresponding operators acting only in the z variables7 eg arm m 7 WM dt and similarly PM and QM Consider the bilinear operator acting on two functions fg in two vari ables Bf9 ZltPhzfQhyg keZ Problem Prove or disprove HBfglls2 S Cllfllsllglls


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