6 problems

Let $0<p<\infty$ and let $\mathbb{X}$ be a quasi-Banach space. Suppose that a vector-valued $L_p(\mathbb{X})$ space admits an averaging projection onto $\mathbb{X}$. Kalton's conje…

Sharp lower endpoint conjecture. The infimum of the exponents $p$ for which this estimate holds is

Let $T$ be the dilated averaging operator considered in Theorem 1, with dilation parameter $\omega\in(0,2)$, and let $(1/p,1/q)$ denote the reciprocal exponent pair for its…

Finite-type Sobolev improving conjecture. The operator $A_{\gamma}$ maps $L^p$ to $L^p_{\alpha}$ for all $p\geq 2$ and

Non-degenerate curve Sobolev improving conjecture. If $\gamma$ is non-degenerate and $2n-2<p<\infty$, then

Let $A$ be the averaging operator considered in Theorem 1.1, and let $I$ be the parameter interval. Assume that the condition denoted by (πLfold) holds for every $t\in I$, with no…