Andrew D. Bailey | Search arXiv

arXiv2012

Boundedness of Maximal Operators of Schrödinger Type with Complex Time

Results of P. Sjölin and F. Soria on the Schrödinger maximal operator with complex-valued time are improved by determining up to the endpoint the sharp $s \geq 0$ for which boundedness from the Sobolev space $H^s(\mathbb{R})$ into $L^2(\mathbb{R})$ occurs. Bounds are established for not only the Schrödinger maximal operator, but further for a general class of maximal operators corresponding to solution operators for certain dispersive PDEs. As a consequence of additional bounds on these maximal operators from $H^s(\mathbb{R})$ into $L^2([-1, 1])$, sharp results on the pointwise almost everywhere convergence of the solutions of these PDEs to their initial data are determined.

math.AP↗

arXiv2011

Pointwise Convergence of Dyadic Partial Sums of Almost Periodic Fourier Series

It is a classical result that dyadic partial sums of the Fourier series of functions $f \in L^p(\mathbb{T})$ converge almost everywhere for $p \in (1, \infty)$. In 1968, E. A. Bredihina established an analogous result for functions belonging to the Stepanov space of almost periodic functions $S^2$ whose Fourier exponents satisfy a natural separation condition. Here, the maximal operator corresponding to dyadic partial summation of almost periodic Fourier series is bounded on the Stepanov spaces $S^{2^k}$, $k \in \mathbb{N}$ for functions satisfying the same condition; Bredihina's result follows as a consequence. In the process of establishing these bounds, some general results are obtained which will facilitate further work on operator bounds and convergence issues in Stepanov spaces. These include a boundedness theorem for the Hilbert transform and a theorem of Littlewood--Paley type. An improvement of "$S^{2^k}$, $k \in \mathbb{N}$" to "$S^p$, $p \in (1, \infty)$" is also seen to follow from a natural conjecture on the boundedness of the Hilbert transform.

math.CA↗