APDE seminar @EHU: Sharp L^2 Bounds for Maximal Functions Along Arbitrary Directions

Abstract

We study sharp L^2-bounds for the maximal function along N arbitrary directions in R^n. Although subcritical on the Kakeya scale, the problem hides a critical multiscale Kakeya phenomenon for directions lying on algebraic curves. Indeed, using polynomial partitioning, the operator decomposes into cell and boundary terms. The main difficulty lies in the boundary term, corresponding to directions constrained to an algebraic curve of bounded degree.

We address this by successively decomposing the curve into pieces with doubling, and then roughly constant, curvature. For these pieces, we establish an almost orthogonality principle by projecting the directions onto curves in lower-dimensional affine spaces. The argument combines tools from algebraic and differential geometry with ideas from approximation theory (Chebyshev systems).

This talk reports on joint work with F. Di Plinio (Università di Napoli Federico II)