Variants of the Maximal Double Hilbert Transform

Abstract

ABSTRACT. We prove the boundedness on LP(T2), 1 < p < oo, of two variants of the double Hubert transform and maximal double Hubert transform. They have an application to a problem of almost everywhere convergence of double Fourier series.

Uniform estimates for families of singular integrals and double Fourier series

“…where H x , denotes the Hilbert transform in x' (actually a variant of it since the kernel is smooth at \x'\ = 2^2"*' just as it is for C y , [8]). So WG^p < c p , 1 < p < oo.…”

“…This is Theorem 3 and Theorem 4 of [8]. Now we consider the operator Tf(x, y) defined as in (1) In other words, if we add up the T q 's corresponding to all pairs we are left with, the resulting operator will have a convolution kernel depending upon the height y at which the operator is evaluated.…”

𝐿² boundedness of highly oscillatory integrals on product domains

“…The existence of the limit for the operator J2 <¡>k(x')<¡>h(y')*f(x,y) (k,h)€Bv has been discussed in [4] and [5]. Since N(y) takes on only a finite number of values the same holds for our operator T. We are going to prove the following theorem.…”

“…where D Ç R x R is symmetric with respect to the origin (more precisely the cut off of the domain of integration given by \d was smooth [4]) but not necessarily a rectangle, otherwise Hi would simply be the double Hubert transform.…”

“…The operator is then (see [4,5]) (2) H2f(x,y) = II -J-1f(x -x',y-y')dy'dx', Dy where nothing is assumed on the dependency upon y of Dy. One is led to this operator by the study of a problem of almost everywhere convergence of double Foureir series (see [6] where the partial sums operator is precisely S^n2 ) or, in other words, by the study of the boundedness on LP(T x T) of the following operator…”