Research Problems in Function Theory — Problem 7.63

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\ge2$, $\alpha>0$ and $R>0$, the Bochner--Riesz operator on Schwartz functions is defined by \[\widehat{T_R^\alpha f}(\xi)=\big(1-|\xi|^2/R^2\big)_+^\alpha\hat f(\xi), \qquad\hat f(\xi)=\int_{\mathbb R^n}f(x)e^{-2\pi i x\cdot\xi}\,dx, \] where $s_+=\max(s,0)$ and Schwartz functions are smooth functions all of whose derivatives decay faster than every inverse power. For $f\in L^p$, $p=2n/(n+1)$, interpret $T_R^\alpha f=K_R^\alpha*f$, with $K_R^\alpha$ the inverse Fourier transform of the displayed multiplier. The kernel belongs to the conjugate Lebesgue class, so this defines the convolution for such $f$.

Statement recorded in the supplied source.

For every $f\in L^{2n/(n+1)}(\mathbb R^n)$ and every $\alpha>0$, does $T_R^\alpha f(x)\to f(x)$ for almost every $x$ as $R\to\infty$? Retain also the source's associated norm-convergence question in dimensions $n\ge3$: does $\|T_R^\alpha f-f\|_{2n/(n+1)}\to0$ for every such $f$ and $\alpha$? The original update reports an affirmative almost-everywhere result in all dimensions by Carbery, Rubio de Francia and Vega. [S2]

Short English statement

Historical question: do these Fourier summation means converge almost everywhere at the specified Lebesgue exponent, and what is the associated norm-convergence behaviour in higher dimensions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307063 (AMR-022-7063). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.63, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.