Olli Saari

Regularity of maximal functions on Hardy-Sobolev spaces: REGULARITY OF MAXIMAL FUNCTIONS ON HARDY-SOBOLEV SPACES

We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy-Sobolev spaces

Parabolic weighted norm inequalities and partial differential equations

\dot{H}^{1,p}(\mathbb{R}^d)

Nonlocal self-improving properties: a functional analytic approach

when

Parabolic BMO and the forward-in-time maximal operator

1/p < 1+1/d

Sharp inequalities for one-sided Muckenhoupt weights

. This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy-Sobolev spaces

Characterizations of interior polar sets for the degenerate p-parabolic equation

\dot{h}^{1,p}(\mathbb{R}^d)

Homeomorphisms of the Heisenberg group preserving BMO

in the same range of exponents.

On weights satisfying parabolic Muckenhoupt conditions

We investigate parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to nonlinear parabolic partial differential equations. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO from Mosers seminal paper through parabolic Muckenhoupt weights and maximal functions.

On regularity of weak solutions to parabolic systems

A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi–Mingione–Sire and Bass–Ren. Its flexibility is demonstrated by new applications to non-autonomous parabolic equations with non-local elliptic part and questions related to maximal regularity.

Non-local Gehring lemmas

We study if the parabolic forward-in-time maximal operator is bounded on parabolic