On estimation of the strong maximal functions
Let \[ M_{\text{st}}f(x):=\sup_{I\ni x}\frac 1{| I| }\int_I| f| \quad\text{and}\quad f_{\text{st}}^{\#}(x):=\sup_{I\ni x}\frac 1{| I| }\int_I| f-f_I| , \] where \(I=\{x\in\mathbb R^n; a_i\leq x_i\leq a_i+h, i=1,2,\dots,N\}\) (with \(h_i>0\), \(i=1,2,\dots,N\)) and \(f_I:=\frac 1{| I| }\int_If\), be the strong maximal function and the Fefferman-Stein function, respectively. For \(f\in L^p(\mathbb R^n)\), \(1\leq p<\infty\), partial modulus of continuity with respect to a variable \(x_j\), \(1\leq j\leq N\), is defined by \[ \omega_p^{(j)}(f;\delta)=\sup_{0\leq h\leq\delta}\Big(\int_{\mathbb R^n}| f(x)-f(x_1,\dots,x_j+h_j,\dots,x_N)| ^p dx\Big)^{1/p}\quad(0\leq\delta<\infty) \] and its mean modulus by \[ \overline{\omega}_p(f;\delta):=\inf_{\delta_1\cdots\delta_N=\delta}\max_{1\leq j\leq N}\omega_p^{(j)}(f;\delta_j). \] The main result of the paper is the \(p=1\) estimate \[ (f_{\text{st}}^{\#})^*(t) \leq 2(M_{\text{st}}f)^*(t)\leq c_N\frac 1t\int_0^t\frac{\overline{\omega}_1(f;\tau)}\tau\log^{N-2}\frac t\tau d\tau,\quad 0<t<\infty. \] The author also shows that this estimate is sharp, constructing a function \(f\) such that for all \(t>0\) \[ (f_{\text{st}}^{\#})^*(t)\geq \frac 1t\int_0^t\frac{\overline{\omega}_1(f;\tau)}\tau\log^{N-2}\frac t\tau d\tau \] (as usual \(g^*\) denotes the non-increasing rearrangement of \(g\), i.e., \(g^*(t)=\inf\{\lambda>0; | \{x\in \mathbb R^N; | g(x)| >\lambda\}| \leq t\}\)). This result is an extension of that by \textit{V. I. Kolyada} [Math. Nachr. 133, 43-61 (1987; Zbl 0702.41052)] for \(1<p<\infty\).
scientific article; zbMATH DE number 1175184