Integrability properties of maximal convolution operators
Let \(\{K_j\}\) be a sequence in \(L^1({\mathbb R}^n)\) (or in \( L^1({\mathbb T}^n)\)) and \( K^*f(x):=\sup_{j}| f*K_j(x)| \) the corresponding maximal operator. The main result of the paper under review states that, if there exists \(C>0\) and some \(m\) such that, for all \(1<p\leq 2\), \[ \lambda| \{K^*\chi_A>\lambda\}| ^{1/p}\leq \left(\frac{C}{p-1}\right)^m| A| ^{1/p} \] for every \(\lambda >0\) and every measurable set \(A\), then for \(R>2\) \[ \int_{| x| \leq R}\Phi(K^*f(x))dx\leq c(\log R)^{m+1}\left\{1+\| f\| _{L(\Psi_m)}\right\}. \] Here \(\| f\| _{L(\Psi_m)}:=\int_0^{\infty}f^*(s)\{1+\log^+(1/s)\}^m \{1+\log^+\log^+\log^+(1/s)\}ds\) and \( \Phi(t):=t(1+\log^+t)^{-1}(1+\log^+\log^+et)^{-1}\). In particular, this gives the global integrability result \[ \int (1+| x|){-\epsilon}\Phi(K^*f(x))\, dx \leq C_\epsilon \{ 1+\| f\| _{L(\Psi_m)} \}. \] When \(K^*\) is the Carleson maximal operator corresponding to the partial sums of the Fourier series, this is an extension of an estimate due to \textit{F. Soria} [Rend. Circ. Mat. Palermo (2) 38, No. 3, 371--376 (1989, Zbl 0683.42022)].
scientific article; zbMATH DE number 5013622