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Multidimensional extension of L. C. Young’s inequality. (English) Zbl 0997.26007
Let \(f\) be a complex-valued function on \([0,1]^2\), and for a partition \(\{x_i\}_{i=0}^n\times\{y_j\}_{j=0}^m\) of \([0,1]^2\), let \(\Delta_{ij}\pi(f) :=f(x_i,y_j)-f(x_{i-1},y_j)-f(x_i,y_{j-1})+f(x_{i-1},y_{i-1})\). For \(p,q\geq 1\), let \(RV_{(p,q)}(f)^q\) be the supremum of the sums \(\sum_{j=1}^m(\sum_{i=1}^n|\Delta_{i,j}\pi (f)|^p)^{q/p}\) with respect to all partitions of \([0,1]^2\). Similarly, \(LV_{(p,q)}(f)\) is defined by interchanging the order of the two sums. Let \(RW_{(p,q)}\) be the class of all functions \(f\colon [0,1]^2\mapsto C\) such that \(\|f\|_{RW_{(p,q)}}:= RV_{(p,q)}(f)+V_q(f(\cdot,0))+V_q(f(0,\cdot))+|f(0,0)|<\infty\), where \(V_p(g)\) is the usual \(p\)-variation of a function \(g\) on \([0,1]\). The class \(LW_{(p,q)}\) is defined by symmetry. For \(\bar{x}=(x_1,x_2)\) and \(\bar{y}=(y_1,y_2)\) both in \([0,1]^2\), let \(d(\bar{x},\bar{y}):=\max\{|x_1-y_1|,|x_2-y_2|\}\) and \(\Delta_{\bar{y}}f(\bar{x}) :=f(x_1,x_2)-f(x_1,y_2)-f(y_1,x_2)+f(y_1,y_2)\). If \(\lim_{\delta\to 0}\sup\{\Delta_{\bar{y}}f(\bar{x})\colon d(\bar{x},\bar{y})<\delta\}=0\) then it is said that \(f\) has a jump at \(\bar{x}\). Let \(f\in RW_{(p_1,p_2)}\), \(RV_{(q_1,q_2)}(g)<\infty\), and for \(i=1,2\), \((1/p_i)+(1/q_i)>1\). The author proved that if \(f\) and \(g\) do not have any common jump points then there exists a constant \(c\) such that the Riemann-Stieltjes integral \(\smallint_{[0,1]^2}f dg\) exists, and its absolute value does not exceed the bound \(c\|f\|_{RW_{(p_1,p_2)}}RV_{(q_1,q_2)}(g)\). The statement also holds when \(RW\) and \(RV\) are replaced with \(LW\) and \(LV\) throughout. This theorem when \(p=q\) is generalized to functions defined on \([0,1]^n\) for an integer \(n\).

MSC:
26B15 Integration of real functions of several variables: length, area, volume
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
28A35 Measures and integrals in product spaces
28A25 Integration with respect to measures and other set functions
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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