Seeking a proof of Xie’s inequality: analogues for series and Fourier series

Abstract

Seeking to free the existence and regularity theory for the Navier–Stokes equations from assumptions about the regularity of a fluid’s boundary, we continue efforts of Wenzheng Xie and myself to prove a certain domain independent inequality for solutions of the steady Stokes equations. For the Laplacian, Xie proved an analogue of the desired inequality by using the maximum principle in obtaining an intermediary result. His conjecture that an analogue of this intermediary result is also valid for the Stokes equations remains unproven. My efforts to circumvent the need for it have led, so far, only to further interesting conjectures. Here, we seek to better understand both Xie’s arguments and mine by applying them to simpler problems concerning series and Fourier series. First, a bound is proven for a series of real numbers that can be interpreted as a bound for the sup-norm of a Fourier cosine series, in terms of the $L^{2}$ -norms of its fractional-order derivatives of orders 1/3 and 2/3. This is generalized to a bound for a weighted sum of a sequence of real numbers. We conjecture that the hypotheses concerning the weights are satisfied by the sequence of numbers $\{ \sin ny\}$, for any nonzero $y\in (-\pi ,\pi )$. If so, we obtain an inequality for the sup-norm of a Fourier sine series, similar to that for a cosine series. Remarkably, the hypotheses for the weights are analogous to those we have been seeking to verify in trying to prove the original inequality for the Stokes equations. We conclude with a remark showing that Xie’s central argument provides a possibly new, very straightforward, proof of Hölder’s inequality for series.