Integrability of double lacunary sine series

Abstract:

We consider mainly the series $\sum {\sum {{a_{jk}}\sin {2^j}x\sin {2^k}y} }$, which converges to a finite function $f(x,y)$ a.e. if $\sum {\sum {a_{jk}^2 < \infty } }$. We show that the (Lebesgue) integrability of ${x^{ - 1}}{y^{ - 1}}f(x,y)$ over the two-dimensional torus is essentially controlled by the quantity $\sum \nolimits _{m = 1}^\infty {\sum \nolimits _{n = 1}^\infty {{{\left ( {\sum \nolimits _{j = m}^\infty {\sum \nolimits _{k = n}^\infty {a_{jk}^2} } } \right )}^{1/2}}} }$. Our result is an extension of the corresponding one by M. C. Weiss [4] from one-dimensional to two-dimensional lacunary sine series.