13/05/2016, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Universidade Nova de Lisboa
On convolution type operator with piecewise continuous symbols on variable Lebesgue spaces
In this talk we discuss three results on convolution type operators acting on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\) under mild assumptions on variable exponents \(p:\mathbb{R}\to(1,\infty)\) : we assume only that the exponent \(p:\mathbb{R}\to(1,\infty)\) is bounded away from one and infinity, and is such that the Hardy-Littlewood maximal operator is bounded on \(L^{p(\cdot)}(\mathbb{R})\). The first result says that the set of the Fourier multipliers \(\mathcal{M}_{L^{p(\cdot)}}(\mathbb{R})\) on the space \(L^{p(\cdot)}(\mathbb{R})\) forms a Banach algebra. The second result is the generalization of the Stechkin inequality for Fourier multipliers on variable Lebesgue spaces saying that the algebra \(V(\mathbb{R})\) of functions \(a\in L^\infty(\mathbb{R})\) of finite total variation \(V(a)\) is continuously embedded into \(\mathcal{M}_{L^p(\cdot)}(\mathbb{R})\), that is, \[\|a\|_{\mathcal{M}_{L^{p(\cdot)}(\mathbb{R})}}\le{\rm const}(\|a\|_\infty+V(a))\quad\mbox{for all}\quad a\in V(\mathbb{R}).\] Now let \(C\) denote the set of all continuous functions on the one-point compactification of \(\mathbb{R}\) and let \(PC\) be the set of all piecewise continuous functions on \(\mathbb{R}\). The completeness of \(\mathcal{M}_{L^{p(\cdot)}}(\mathbb{R})\) and the Stechkin inequality allows us to define the classes \(C_{p(\cdot)}\) and \(PC_{p(\cdot)}\) of continuous and piecewise continuous Fourier multipliers as the closure of \(C\cap V(\mathbb{R})\) and \(PC\cap V(\mathbb{R})\) with respect to the norm of \(\mathcal{M}_{L^{p(\cdot)}}(\mathbb{R})\). The third result concerns the compactness of the commutator \[[aI,W^0(b)]=aW^0(b)-W^0(b)aI\] of the Fourier convolution operator \(W^0(b)\) and the operator of multiplication \(aI\) by a function \(a\) on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\). This result is proved under the assumption that \((a,b)\in (C,PC_{p(\cdot)})\) or \((a,b)\in (PC,C_{p(\cdot)})\) and generalizes the corresponding result by Roland Duduchava proved for constant exponents in 1970's.
