arXiv subjects

Benjamin W. Kosmala

Publications and source records attributed to Benjamin W. Kosmala.

3 recordsLinked to original sources

arXiv2026

Singular solutions to parabolic equations in one space dimension

We construct singular weak solutions to scalar, linear, uniformly parabolic equations with complex coefficients in one space dimension. Our examples show that the space-time integrability provided by the energy estimates is sharp: for every $p>6$, there exists such an equation admitting an energy solution that fails to be $p$-integrable on a compact subset of the space-time domain. In particular, the local boundedness of weak solutions from the De Giorgi--Nash--Moser theory fails for complex coefficients already in one space dimension.

math.AP↗

arXiv2026

$\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$

We establish the first results on $\mathrm{L}^p$ bounds for Riesz transforms associated with non-autonomous second order parabolic differential operators in divergence form with bounded coefficients that depend measurably on all variables. In the case of complex coefficients, we identify the maximal open range of exponents $1<p \leq2$ through the availability of $\mathrm{L}^p$ resolvent bounds. This open range always contains the lower parabolic Sobolev conjugate of $2$ and the result is sharp in spatial dimension $n \geq 2$. For real coefficients, we prove extrapolation to the full range. Our argument relies on novel space-time off-diagonal bounds based on two complementary geometries: parabolic cubes on small scales and regions modeled after the half-order time derivative of a parabolic Bessel potential on large scales.

math.CA↗

arXiv2024

A Note on Complex Interpolation of Quasi-Banach Function Spaces

Kalton and Mitrea characterized complex interpolation spaces of quasi-Banach function spaces as Calderón products if both interpolants are separable. We show that one separability assumption may be omitted and establish a Wolff-reiteration result with one non-separable endpoint space.

math.FA↗