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Lenka Slavíková

Publications and source records attributed to Lenka Slavíková.

26 records · Page 2Linked to original sources
arXiv2018

$L^2\times L^2 \to L^1$ boundedness criteria

We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this function; precisely we show that boundedness holds if and only if $q<4$. We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal $L^2\times L^2\to L^1$ boundedness criterion for bilinear operators associated with multipliers with $L^\infty$ derivatives. This result provides the main tool in the proof of the first theorem and is also manifested in terms of the $L^q$ integrability of the multiplier. The optimal range is $q<4$ which, in the absence of Plancherel's identity on $L^1$, should be compared to $q=\infty$ in the classical $L^2\to L^2$ boundedness for linear multiplier operators.

math.CA↗
arXiv2017

A sharp version of the Hörmander multiplier theorem

We provide an improvement of the Hörmander multiplier theorem in which the Sobolev space $L^r_s(\mathbb R^n)$ with integrability index $r$ and smoothness index $s>n/r$ is replaced by the Sobolev space with smoothness $s$ built upon the Lorentz space $L^{n/s,1}(\mathbb R^n)$.

math.CA↗
arXiv2015

Banach algebras of weakly differentiable functions

The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also established. The special cases of Orlicz-Sobolev and Lorentz-Sobolev spaces are discussed in detail. New results for classical Sobolev spaces on possibly irregular domains follow as well.

math.FA↗
arXiv2015

Norms supporting the Lebesgue differentiation theorem

A version of the Lebesgue differentiation theorem is offered, where the $L^p$ norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue's theorem holds are characterized.

math.FA↗
arXiv2015

On the necessity of bump conditions for the two-weighted maximal inequality

We study the necessity of bump conditions for the boundedness of the Hardy-Littlewood maximal operator $M$ from $L^p(v)$ into $L^p(w)$, where $1<p<\infty$. The conditions in question are obtained by replacing the average of $σ=v^{-\frac{1}{p-1}}$ in the Muckenhoupt $A_p$-condition by an average with respect to certain Banach function space, and are known to be sufficient for the two-weighted maximal inequality. We show that these conditions are in general not necessary for the boundedness of $M$ from $L^p(v)$ into $L^p(w)$.

math.CA↗
arXiv2013

Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in $\rn$, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.

math.FA↗
arXiv2013

Compactness of higher-order Sobolev embeddings

We study higher-order compact Sobolev embeddings on a domain $Ω\subseteq \mathbb R^n$ endowed with a probability measure $ν$ and satisfying certain isoperimetric inequality. Given $m\in \mathbb N$, we present a condition on a pair of rearrangement-invariant spaces $X(Ω,ν)$ and $Y(Ω,ν)$ which suffices to guarantee a compact embedding of the Sobolev space $V^mX(Ω,ν)$ into $Y(Ω,ν)$. The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of $(Ω,ν)$. We then apply this result to the characterization of higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, whose standard example is the Gauss space.

math.FA↗