On the multi-dimensional duality principle of Sawyer type (Q5936963)

Let \(v(\mathbf x)= v(x_1,\dots,x_n)\) be a weight on \(\mathbb R^n_+\). We write \(v\in D\) if there are constants \(c_1,c_2 > 0\) such that \[ \int_{(\mathbf 0,2\mathbf R)} v(\mathbf x) d\mathbf x\leq c_1 \int_{(\mathbf 0, \mathbf R)} v(\mathbf x) d{\mathbf x} \tag{1} \] and \[ \int_{(\mathbf 0,\mathbf R)}v(\mathbf x) d\mathbf x\leq c_2 \int_{(\mathbf R/2, \mathbf R)} v(\mathbf x) d\mathbf x\tag{2} \] for all \(\mathbf R=(R_1,\dots,R_n)\in\mathbb R^n_+\), where \((\mathbf{0},\mathbf R) = (0,R_1)\times \dots \times(0,R_n)\), \((\mathbf 0,2\mathbf{R}) = (0,2R_1)\times \dots \times (0,2R_n)\) and \((\mathbf R/2, \mathbf R) = (R_1/2,R_1)\times \dots \times (R_n/2, R_n)\). We put \((\mathbf 0,\infty)=\) \((0,\infty)^n\). The main result of the paper reads as follows: Let \(0<q<p<\infty\), \(1/r = 1/q - 1/p\) and let \(u,v\) be weights on \(\mathbb R^n_+\) with \(v\in D\). Then \[ \Big(\int_{(\mathbf 0, \infty)} f^q (\mathbf x) u(\mathbf x) d\mathbf x\Big)^{1/q} \leq c_1^{\frac 2{q}} c_2^{\frac{1}{q}-\frac 1{p}} [A_v(u)]^{\frac 1{r}} \Big(\int_{(\mathbf{0},\infty)} f^p (\mathbf x)v(\mathbf x) d\mathbf x\Big)^{1/p} \tag{3} \] for all non-negative functions \(f=f(\mathbf x) = f(x_1,\dots,x_n)\) which are non-increasing in each variable, where \[ A_v (u) = \int_{(\mathbf 0,\infty)} \Big[\Big(\int_{(\mathbf 0,\mathbf x)} u(\mathbf y) d\mathbf y\Big)^{1/q} \Big(\int_{(\mathbf 0,\mathbf x)} v(\mathbf y) d\mathbf y \Big)^{-1/q}\Big]^r v(\mathbf x) d\mathbf x \] and \(c_1,c_2\) are the constants from (1) and (2). On putting \(q=1\), we get \(r=p'\) and inequality (3) turns to the Hölder inequality for non-negative functions \(f=f(x_1,\dots,x_n)\) which are non-increasing in each variable. When the weight \(v\) satisfies \[ v(\mathbf x) = v_1(x_1) \times \dots \times v_n(x_n),\tag{4} \] that is, \(v\) is the product of one-dimensional weights, then the results mentioned above coincide with that of \textit{S. Barza} [``Weighted multi-dimensional integral inequalities and applications'', Doctoral Thesis, Luleå University of Technology, Sweden (1999)]. Note that the author shows that there are \(v\in D\) which are not of the form~(4).