30 problems

Let $v$ be a solution of the forced Navier–Stokes equations on $\mathbb{R}^3$, and suppose that $v$ first loses smoothness at time $T^*>0$. For a fixed constant $c>0$, consider the…

Let $v$ be a weight, let $Q$ be a non-negative definite symmetric matrix, and suppose that the Sobolev hypothesis holds for some $sigma>1$. Define … For data $f\in L^A(v;\Omega)$,…

Strict rescaling quotient conjecture. One has

Let $1<p<\infty$, let $(\mu,\nu)$ be a pair of weights, and let $\Phi$ and $\Psi$ be Young functions with complementary functions $\bar{\Phi}$ and $\bar{\Psi}$. For a cube $Q$, wri…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, let $1<p<\infty$, and suppose that $Q$ and $v$ satisfy $v\in L^1(\Omega)$ and $|Q|_{\mathrm{op}}^{p/2}\leq kv$. Assume that…

Kolwicz–Leśnik–Maligranda conjecture. If $X\neq L_{\infty}[0,1]$, then

A Schauder-Orlicz decomposition is a Schauder decomposition whose associated coefficient space is an Orlicz sequence space. Fix the dimensions of its component subspaces. Uniquenes…

Extension conjecture. The theory developed in the first part of the thesis should extend to the case in which $\phi$ grows linearly, with an additional non-integral term in the ene…

Let $M_1,M_2\colon \mathbb{R}\to[0,\infty)$ be the two Orlicz functions defining the intersected Orlicz balls, and let $R$ be the parameter governing their scale. As in the case of…

The affine Orlicz Pólya–Szegő principle concerns the comparison between an Orlicz–Sobolev function and its symmetric decreasing rearrangement, with the relevant affine-invariant Or…

Improvement conjecture. The statement of the continuity theorem should remain valid under the weaker condition $p^->n$. This concerns weak convergence of the corresponding solution…

Let $G$ be a bounded open subset of $\mathbb{R}^n$, let $\Phi$ be an Orlicz function, and let $W_0^{1,\Phi}(G)$ be the subspace of the first-order Orlicz-Sobolev space consisting o…

Let $\Phi:[0,\infty)\to[0,\infty)$ be a continuous increasing function with $\Phi(0)=0$, and suppose that … For $f\in L\log L(\mathbb{T}^2)$, write $S_{ij}(x_1,x_2,f)$ for the rect…

Noncommutative Musielak–Orlicz manifold conjecture. A similar construction can be carried out in the noncommutative case.

Jenčová's extension conjecture. Jenčová's smooth manifold construction can be extended from $\N^+_{\star01}$ to $\N^+_{\star0}$.

Let $\Phi$ be an Orlicz function satisfying the $\triangle_2$-condition, and let $x=(x_n)_{n\geq 1}$ be a noncommutative $L_1$-bounded martingale. For a martingale $u$, write…

Let $\Phi$ be an Orlicz function with Boyd indexes $p_{\Phi}$ and $q_{\Phi}$. Consider the noncommutative Khintchine inequality in the range $q_{\Phi}<2$ for finite sequences in th…

The relevant Orlicz spaces are the pair $\bigl(L^{\cosh-1},L\log(L+1)\bigr)$, which is proposed as a framework for Statistical Physics in both the classical and quantum settings. D…

Let $I_\alpha$ be the fractional integral operator, let $u$ be a weight on $\mathbb R^n$, and let $M_{\Phi,\alpha}$ be the Orlicz fractional maximal operator … where…

Exponential Orlicz discrepancy conjecture. The estimate above should hold for suitable $N$-point sets in every dimension $d\geq 3$. This conjecture predicts an improvement over the…

Orlicz endpoint regularity conjecture. In this endpoint case,

Uniqueness conjecture. If there is a unique (up to equivalence near $0$) mean zero function $f$ whose independent copies are equivalent in $L_p$ to the unit vector basis in $l_M$,…

Let $u$ and $v$ be weights, let $1<p<\infty$, and consider the two-weight $A_p$ condition … Here $\mathcal Q$ denotes the relevant family of cubes and $p'=p/(p-1)$. Orlicz bump con…

The $L\log L$ conjecture. A natural reformulation of the almost-everywhere Fourier-convergence conjecture is that condition $$ holds for ${mathcal C}$ with

Fourier series are understood in their usual partial-sum sense, and $L\log L$ denotes the Orlicz class associated with the growth function $t\mapsto t\log({\rm e}+t)$. The…