We present two new general \(L^p\)-integral inequalities involving the primitive of a function. They extend the applicability of some established results in the literature, mainly thanks to the presence of an intermediate function that can be adapted to the context. We thus contribute to the understanding of the interaction between the \(L^p\)-integrals of functions and their primitives.
The classic Besicovitch projection theorem asserts that if a set is purely 1-unrectifiable with finite length in ℝ^2, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely 1-unrectifiable. We extend the two-projection theorem to certain families nonlinear projections and consider applications to pinned distance sets, radial projections, and curve projection operators. Further, we use a multiscale framework to obtain a quantitative version of our nonlinear two-projection theorem. Our arguments utilize methods introduced by Tao, who provided a quantitative treatment of the classic linear two-projection theorem.
We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.
Let U be an open set in ℝ^d. A continuous function f U →ℝ is strongly nowhere differentiable if and only if for each γ∈(0,1] and for each unit speed C^1,γ curve c [a,b] → U, the composition f∘ c [a,b] →ℝ is nowhere differentiable on (a,b). For bounded U, let U be the closure of U and C(U) be the Banach space of continuous real-valued functions on U with the sup norm. Theorem. In the sense of the Baire category theorem, almost every f∈ C(U) is strongly nowhere differentiable on U.
