58 problems

Trace singularity-spectrum conjecture. Theorem $$ holds for every $r\in\mathbb{R}$ simultaneously on a prevalent set.

Frisch–Parisi conjecture. In that Baire functional space, any typical element $f$ obeys some multifractal formalism and satisfies

Let $L(p,\theta,q)$ denote the spaces introduced in the preceding interpolation theorem, and let $[\,\ ]_\sigma$ denote the complex interpolation functor. For $0<p_0<1$ and…

Let $H=-\Delta+V$ be the Schrödinger operator with the barrier potential $V$ defined in the paper, and let $H_0=-\Delta$ be the free Schrödinger operator. For parameters $p$ and…

Theorem concerns characterizations of alpha-modulation spaces, with parameter $\alpha$ and exponent $p$; the relevant inhomogeneous Besov space is denoted by…

Let $1\leq p,s<\infty$. For some $\mu>0$, let $C_\mu$ be a constant such that, for every $j\geq1$ and every sequence $\{f_k\}$ satisfying…

Consider the one-dimensional hypo-viscous compressible Navier–Stokes equations with viscosity exponent $\beta=\frac{1}{2}$. Here $B_{2,1}^{\frac{1}{2}}$ denotes the critical Besov…

Classification-rate-optimality conjecture. The bounds in and remain rate-optimal, including for values of $p$ and $q$ other than $\infty$, analogously to the regression setting.

Let $u_0\in L^2$ be initial data, and consider sequences of Leray–Hopf weak solutions of the Navier–Stokes equations whose inviscid limits are denoted by $u$. The critical Besov cl…

Prevalent singularity spectrum conjecture. For every $f\in\widetilde{b}^{\mu,\Psi}_{p,q}([0,1]^d)$,

Besov-space extension conjecture. For each $p\in(2,3)$, there is a range of Besov spaces close to $L^{p,\infty}$ for which a version of Theorem $$ can be established.

Triebel's composition conjecture. If $s>1+\frac{1}{p}$, this map is bounded if and only if

Let $(u,b)$ be a weak solution of the ideal MHD system, and define the total energy and cross helicity by … … Here $C_{x,t}^\alpha$ denotes joint Hölder regularity and…

Let $A^{s,\tau}_{p,q}(\mathbb{R}^n)$ denote either a Besov-type or Triebel--Lizorkin-type space, with $s>0$, $p,q\in[1,\infty]$ (and $p<\infty$ in the Triebel--Lizorkin case), and…

Peetre's conjecture. Determine the real interpolation space

Let $B^{\boldsymbol{r}}_{\infty,\theta}$ be the Nikol'skii–Besov class of periodic functions with mixed smoothness, let $L_\infty$ be the supremum-norm space, let $B_{\infty,1}$ be…

Let $B^{\boldsymbol{r}}_{\infty,\theta}$ be the Nikol'skii–Besov class of periodic functions with mixed smoothness, let $Q_n^{\boldsymbol{\gamma}}$ denote the relevant hyperbolic-c…

Norm-equivalence conjecture. For $1\leq p,q\leq\infty$, this Besov quasi-norm is equivalent to the norm in Definition 3.1 of XXY.

Lee–Rogers–Seeger local space-time conjecture. The estimate

The spaces $BV^q$ and $BV$ are being compared through their jump variations: for $u\in BV(\mathbb{R}^N,\mathbb{R}^d)$, the jump part of its distributional derivative has total vari…

Let $B^s_{p,q}(\mathbb{R}^d_\theta)$ denote the inhomogeneous Besov space defined through the Littlewood–Paley decomposition, and let the quantum Besov spaces introduced by L. Lafl…

Let $u$ be a classical axisymmetric solution of the axisymmetric Navier–Stokes equations, and let a critical Besov norm replace the weak-$L^3$ norm appearing in Theorem. Critical B…

Let $-n<\varrho<0$, $0<p<\infty$, and $0<q\leq\infty$. Consider the spaces ${\Lambda}_{\varrho}B^s_{p,q}({\mathbb R}^n)$ and ${\Lambda}^{\varrho}B^s_{p,q}({\mathbb R}^n)$, and let…

The paper considers the Besov-type borderline space $b^{2/3}_{3,\infty}$, which is larger than the fractional Sobolev space $W^{2/3,3}$. Borderline-space conjecture. The results of…

Peller's conjecture. If $f\in\dot{B}^{\frac{1}{p}}_{\infty,p}(\mathbb{R})$, then $f$ is $\mathcal{L}_p$-operator Lipschitz.