Logarithmic inequalities for Fourier multipliers
It is known that for any bounded function \(m:\mathbb{R}^{d}\rightarrow\mathbb{C}\) there is a unique bounded operator \(T_{m}\) on \(L^{2}(\mathbb{R}^{d}),\) called the Fourier multiplier with symbol \(m\), which is given by \(\widehat{T_{m}f}^{{}}=m\widehat{f}.\) In this paper we shall consider the following class of symbols. Let \(\vartheta \) be a Levy measure on \(\mathbb{R}^{d}\): that is, a nonnegative Borel measure on \(\mathbb{R}^{d}\) such that \(\vartheta( \{ 0\}) =0\) and \[ \int_{\mathbb{R}^{d}}\min \left\{ \left| x\right| ^{2},1\right\}\,d\vartheta(x) <\infty . \] Assume further that \(\mu \) is a finite Borel measure on the unit sphere \(S\) of \(\mathbb{R}^{d}\) and fix two Borel functions \(\phi \) on \(\mathbb{R}^{d}\) and \(\psi \) on \(S\) which take values in the unit circle of \(\mathbb{C}.\) We define the associated multiplier \(m=m_{\phi ,\psi ,\mu ,\vartheta }\) on \(\mathbb{R}^{d}\) by \[ m\left( \xi \right) =\frac{\frac{1}{2}\int_{S}\left\langle \xi ,\theta \right\rangle ^{2}\psi \left( \theta \right)\,d \mu(\theta) +\int_{\mathbb{R}^{d}}\left[ 1-\cos \left\langle \xi ,x\right\rangle \right] \phi \left( x\right)\,d\vartheta(x) }{\frac{1}{2}\int_{S}\left\langle \xi ,\theta \right\rangle ^{2}\,d\mu(\theta) +\int_{\mathbb{R}^{d}}\left[ 1-\cos \left\langle \xi ,x\right\rangle \right] \,d\vartheta (x) } \tag{1} \] if the denominator is not \(0,\) and \(m\left( \xi \right) =0\) otherwise. Here \( \langle\,\cdot\,,\,\cdot\,\rangle \) is the scalar product in \(\mathbb{R}^{d}.\) Throughout the paper, the functions \(\phi ,\psi :[o,\infty )\rightarrow [ o,\infty )\) are given by \[ \phi \left( x\right) =e^{x}-1-x\quad\text{and}\quad\psi \left( x\right) =\left( x+1\right) \log \left( x+1\right) -x. \] Define the class \(L\log L(\mathbb{R}^{d}) \) by \[ L\log L(\mathbb{R}^{d}) =\left\{ f:\mathbb{R}^{d}\rightarrow\mathbb{C}:\int_{\mathbb{R}^{d}}\psi \left( \left| f\left( x\right) \right| \right)\, dx<\infty \right\} . \] The main result of the paper is in the following. Let \(m=m_{\phi ,\psi ,\mu ,\vartheta }\) be given by \(\left( 1\right) ,\) with \(\mu ,\vartheta ,\phi \) and \(\psi \) satisfying the above assumptions. Then for any \(K>1,\) any \(f\in L\log L(\mathbb{R}^{d}) \) and any Borel subset \(A\) of \(\mathbb{R}^{d}\) we have \[ \int_{A}\left| T_{m}f\left( x\right) \right| \,dx\leq K\int_{\mathbb{R}^{d}}\psi \left( \left| f\left( x\right) \right| \right) \,dx+\frac{ \left| A\right| }{2\left( K-1\right) }. \]
scientific article; zbMATH DE number 6176506
scientific article