Maximum Principle for Intermediate Riesz Kernels

Definitions and mathematical statement

Let $d\geq 3$ be an integer and let $s\in{\mathbb{R}}$ satisfy $1<s<d-1$. Write $|\cdot|$ for the Euclidean norm on ${\mathbb{R}}^d$, and define the vector-valued kernel \[ K_s(x)=\frac{x}{|x|^{s+1}},\qquad x\in{\mathbb{R}}^d\setminus\{0\}. \] For a nonnegative real-valued function $f\in C_c^\infty({\mathbb{R}}^d)$, define \[ T_s f(x)=(K_s*f)(x)=\int_{{\mathbb{R}}^d}\frac{x-y}{|x-y|^{s+1}}f(y)\,\mathrm{d}y, \qquad \operatorname{supp}f=\overline{\{y\in{\mathbb{R}}^d:f(y)\neq0\}}. \] The integral is absolutely convergent because $s<d$ and $f$ is smooth with compact support; the value of the kernel at the origin does not affect it.

The question is whether, for every such pair $(d,s)$, there exists a finite constant $C=C(d,s)>0$, independent of $f$, such that every nonnegative $f\in C_c^\infty({\mathbb{R}}^d)$ satisfies \[ \sup_{x\in{\mathbb{R}}^d}|T_s f(x)| \leq C\sup_{x\in\operatorname{supp}f}|T_s f(x)|. \] For $f\equiv0$, interpret the right-hand supremum as $0$; equivalently, formulate the question for nonzero nonnegative $f$. This convention resolves the empty-support case, which the source does not specify. The dependence $C=C(d,s)$ expresses the interpretation that the parameters are fixed before the constant is chosen; the supplied statement does not explicitly discuss uniformity in these parameters.

The supplied record labels the problem partially solved. Its dated curation reports positive-data results for exponents below one and for exponents at least dimension minus one, an unresolved intermediate range, and failure for signed data. These are supplied literature assessments, not independently verified assertions here. The present question retains the strict range $1<s<d-1$ and the nonnegativity hypothesis; the curation does not supply complete hypotheses for its broader claims or specify the status at $s=1$.

Short English statement

For each fixed dimension $d\geq3$ and exponent $1<s<d-1$, is the largest magnitude of the Riesz convolution of any nonnegative smooth compactly supported function bounded by a constant depending only on $d,s$ times its largest magnitude on the function's support? The supplied partially solved classification requires independent literature verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003536$, title: Maximum Principle for Intermediate Riesz Kernels; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record gives proposed year 2017 and no named proposer. Its statement status is corrected_verified, with a supplied claim of manual semantic, typographic, and LaTeX review; that claim is not independent mathematical verification of this draft.
• Supplied report citation: Real Analysis, Harmonic Analysis, and Applications (2018), Oberwolfach Reports, pp. 2101--2164, DOI 10.4171/owr/2017/34. https://doi.org/10.4171/owr/2017/34. The background attributes the collection to workshop open-problem discussions. The literature-source label identifies the same link as Oberwolfach Report 34/2017; the differing year labels are retained without resolution.
• Eiderman--Nazarov, Maximum principle for the Riesz transform. Supplied literature source: https://arxiv.org/abs/1701.04500.
• Literature checked at: 2026-08-21. This exact date belongs to supplied dated curation derived from the OWR material, not an independently verified literature assessment. The curation labels the problem partially solved and reports positive-data results below exponent one and at exponents at least dimension minus one, an open intermediate range, and failure for signed data. Time-sensitive status and the cited claims require independent verification.