12 problems

Let $p\colon\mathbb{R}\to[1,2]$ be a variable exponent, and let $p'$ be defined pointwise by … For a function $f$ on $\mathbb{R}$, write $\hat f$ for its Fourier transform and…

Variable-exponent conjecture. Theorems concerning multiplicity for … should remain valid when $q$ is replaced by a variable exponent $q$.

Weak compactness conjecture. If $p^{-}=1$, then $T_{\varphi}$ is weakly compact on $L^{p(\cdot)}([0,1])$ if and only if

Let $p(\cdot)$ be a variable exponent on $\mathbb{R}^n$, let $\omega$ be a weight, and let $\mathcal M$ denote the Hardy–Littlewood maximal operator. Write…

Let $p:{\mathbf R}^d\to[1,\infty]$ satisfy $1<p_-\leq p_+<\infty$, let $w$ be a weight, and let $A_{\mathrm{FW}}$ and $A_{\mathrm{strong}}$ denote the relevant Muckenhoupt-type con…

Existence conjecture. The problem has a solution for every $\lambda>0$ under this almost critical assumption.

Let $L^{p(\cdot)}_\mathrm{germ}$ denote the germ space associated with the variable Lebesgue space $L^{p(\cdot)}$, and suppose in particular that $L^\infty\subset L^{p(\cdot)}$. Ge…

For a functional with variable exponents or a double-phase potential, the preceding examples involve an exponent crossing the dimensional threshold. Dimensional-threshold conjectur…

Let $p\funccolon\mathbb{R}^n\to\mathbb{R}$ satisfy the log-Hölder continuity condition, with $p^-\in(1,\infty)$, and let $\{\psi_\epsilon\}_{\epsilon>0}$ be a family of functions a…

Singular integral conjecture. Under these inequalities, the singular integral operator $S$ is bounded on $L^{p(\cdot)}(\Gamma,\varphi_{t,\gamma})$. The conjecture is motivated by t…

Let $\Gamma$ be a simple Carleson curve, let $w:\Gamma\to[0,\infty]$ be a weight, and let $p:\Gamma\to(1,\infty)$ be continuous and satisfy the Dini–Lipschitz condition. Let…

Let $\Gamma$ be a simple rectifiable curve, let $p:\Gamma\to[1,\infty]$ be measurable and finite almost everywhere, and let $w:\Gamma\to[0,\infty]$ be a weight. Suppose that the Ca…