1   What is Fourier Restriction Theory   Main object  f     d      f   x             b   e 2   i x      b         Notation  We will write e   x       i e 2   i x       x     d is a spatial variable  and       d is the frequency variable   The frequencies  or Fourier transform  of f is restricted  to a set R  where R we will always be finite   so no need to worry about convergence issues    Goal  Understand the behaviour of f in terms of properties of R   Example   i  Schr dinger equation   Easy    2   i           u   0  with initial data u   x   0       n   1 N b n e   n x      x   t       x 1   x 2    Then since e   n x   n 2 t   i e     n   n 2       x   t      we might consider R       n   n 2     n   1       N      ii  Dirichlet polynomials   with b n   1  partial sums of Riemann Zeta function  We might consider R     1 2   log n   n   N 2 N   Both avoid linear structure     log n   is a concave set  getting closer and closer together        n   n 2     lie on a parabola    Guiding principle  if properties of an object avoid  linear  structure  then we expect some random or average behaviour   The above examples avoid linear structure using some notion fo curvature  See Bourgain     p   paper    extra behaviour   Square root cancellation  If we add   1  randomly N times  then we expect a quantity with size N 1 2   Theorem 1 1   Khinchin s inequality    Assuming that       n   n   1 N be IID random variables with       n   1           n     1     1 2  1   p      x 1       x N      Then   Notation    p means   but the constant may depend on p   Proof  Without loss of generality  x 1       x n      Without loss of generality    x   2   1   p   2  want to show         n   n x n   2     1   What about general exponents p   The equality here is the Layer cake formula  which is true for any p     0         Let     0  Study the random variable e     n   n x n     0         Fact  1 2 e z   1 2 e   z   e z 2 2  to check  use the Taylor series   So we can get           e     n   n x n             e     n   n x n     Chebyshev s inequality       n e   2   x n   2   2   e   2   2     By symmetry    Choose     e   2   Use in Layer cake   Lower bound  use H lder s inequality  X     n   n x n   1 p   1 q   1     Can you find a more intuitive proof  E mail Dominique Maldague   Corollary            n   1 N   n f n   x     p d x     p       n   1 N   f n   x     2   p   2 d x   Useful for exercises   Return to Fourier restriction context      f   x       n   1 N e   n x   R     1       N   g   x       n   1 N e   n 2 x   R     1 2   2 2       N 2       Both f   g are 1 periodic  So study them on       0   1    f   0     N    f   x       N for     0   c 1 N    g   0   i N    g   x       N for x     0   c 1 N 2     For the first one  N p   1  organised behaviour  dominates as soon as p   2  and for the second one  N p   2 dominates for 2   p   4   square root cancellation behaviour lasts for longer