1. CMB Online first
||A Multiplier Theorem on Anisotropic Hardy Spaces|
We present a multiplier theorem on anisotropic
Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin
condition, we obtain boundedness of the multiplier operator $T_m
: H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$
that depends on the eccentricities of the dilation $A$ and the
level of regularity of a multiplier symbol $m$. This extends
the classical multiplier theorem of Taibleson and Weiss.
Keywords:anisotropic Hardy space, multiplier, Fourier transform
Categories:42B30, 42B25, 42B35
2. CMB 2011 (vol 56 pp. 326)
3. CMB 2011 (vol 54 pp. 544)
4. CMB 2011 (vol 55 pp. 689)
||A Pointwise Estimate for the Fourier Transform and Maxima of a Function|
We show a pointwise estimate for the Fourier
transform on the line involving the number of times the function
changes monotonicity. The contrapositive of the theorem may be used to
find a lower bound to the number of local maxima of a function. We
also show two applications of the theorem. The first is the two weight
problem for the Fourier transform, and the second is estimating the
number of roots of the derivative of a function.
Keywords:Fourier transform, maxima, two weight problem, roots, norm estimates, Dirichlet-Jordan theorem
5. CMB 2010 (vol 54 pp. 159)
||Hardy Inequalities on the Real Line|
We prove that some inequalities, which are considered to be
generalizations of Hardy's inequality on the circle,
can be modified and proved to be true for functions integrable on the real line.
In fact we would like to show that some constructions that were
used to prove the Littlewood conjecture can be used similarly to
produce real Hardy-type inequalities.
This discussion will lead to many questions concerning the
relationship between Hardy-type inequalities on the circle and
those on the real line.
Keywords:Hardy's inequality, inequalities including the Fourier transform and Hardy spaces