Luboš Pick

An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded

Abstract We give an example of a function p such that the Hardy-Littlewood maximal operator is not bounded on the generalized Lebesgue space L p(x) .

Optimal sobolev embeddings

Abstract The aim of this paper is to study Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. We establish the equivalence of a Sobolev imbedding to the boundedness of a certain weighted Hardy operator. This Hardy operator is then used to prove the existence of rearrangement-invariant norms that are optimal in the imbedding inequality. Our approach is to use the methods and principles of Interpolation Theory.

An elementary proof of sharp Sobolev embeddings

We present an elementary unified and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Brezis and Wainger.

Sobolev embeddings into BMO, VMO, andL∞

LetX be a rearrangement-invariant Banach function space onRn and letV1X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV1X is continuously embedded into BMO or intoL∞. In particular, we show thatLn, ∞ is the largest rearrangement-invariant spaceX such thatV1X is continuously embedded into BMO and, similarly,Ln, 1 is the largest rearrangement-invariant spaceX such thatV1X is continuously embedded intoL∞. We further show thatV1X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inLn, ∞. A compact inclusion ofV1X intoC0 is characterized as well.

Are generalized Lorentz “spaces” really spaces?

Let

Duals of Optimal Spaces for the Hardy Averaging Operator

w

Optimality of Function Spaces in Sobolev Embeddings

be a non-negative measurable function on

Weighted estimates for the Hilbert transform of odd functions

(0,\infty)

Two-weight weak and extra-weak type inequalities for the one-sided maximal operator

, non-identically zero, such that

Ridged Domains, Embedding Theorems and Poincaré Inequalities

W(t)=\int_0^tw(s)ds0