Petr Honzík

Maximal transference and summability of multilinear Fourier series

We obtain a maximal transference theorem that relates almost everywhere convergence of multilinear Fourier series to boundedness of maximal multilinear operators. We use this and other recent results on transference and multilinear operators to deduce Lp and almost everywhere summability of certain m-linear Fourier series. We formulate conditions for the convergence of multilinear series and we investigate certain kinds of summation.

On the p-independence boundedness property of Calderón-Zygmund theory

Abstract For 0 ≦ α < 1 we construct examples of even integrable functions Ω on the unit sphere 𝕊 d-1 with mean value zero satisfying such that the L 2-bounded singular integral operator T Ω given by convolution with the distribution p.v. Ω(x/|x|)|x|-d is not bounded on L p (ℝ d ) when . In particular, we construct operators T Ω that are bounded on L p exactly when p = 2.

On the good- inequality for nonlinear potentials

Abstract. This paper concerns an extension of the good-λ inequality for fractional integrals, due to B. Muckenhoupt and R. Wheeden. The classical result is refined in two aspects. Firstly, general nonlinear potentials are considered, and secondly, the constant in the inequality is proven to decay exponentially. As a consequence, the exponential integrability of the gradient of solutions to certain quasilinear elliptic equations is deduced. This in turn is a consequence of certain Morrey space embeddings which extend classical results for the Riesz potential. In addition, the good-λ inequality proved here provides an elementary proof of the result of Jawerth, Perez and Welland regarding the positive cone in certain weighted Triebel-Lizorkin spaces.

On an inequality of Sagher and Zhou concerning Stein's lemma

AbstractWe provide two alternative proofs of the following formulation of Stein’s lemma obtained by Sagher and Zhou [6]: there exists a constant A > 0 such that for any measurable setE⊂ [0, 1], |E| ≠ 0, there is an integerN that depends only onE such that for any square-summable real-valued sequence {ck}k=0/∞ we have:1

On maximal functions for Mikhlin¿Hörmander multipliers

A \cdot \sum\limits_{k > N} {\left| {c_k } \right|} ^2 \leqslant \mathop {sup}\limits_I \mathop {inf}\limits_{a \in \mathbb{R}} \frac{1}{{\left| I \right|}} \int_{I \cap E} {\left| {f(t) - a} \right|^2 } dt,

Dimension of images of subspaces under Sobolev mappings

where the supremum is taken over all dyadic intervals I and

Maximal Marcinkiewicz multipliers

f(t) = \sum\limits_{k = 0}^\infty {c_k r_k } (t),